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November 2005

Dr. John W. Labadie Department of Civil Engineering

Colorado State University Ft. Collins, CO 80523 Ph. 01-970 491-6898

Email: labadie@engr.colostate.edu

MODSIM: River Basin Management Decision Support System

Version 8.0

contact:

DRAFT

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MODSIM 8.0: River Basin Management Decision Support System

I. INTRODUCTION

MODSIM 8.0 is a generalized river basin management decision support system (DSS) designed as a computer-aided tool for developing improved basin wide and regional strategies for short-term water management, long-term operational planning, drought contingency planning, water rights analysis and resolving conflicts between urban, agricultural, and environmental concerns. Sprague and Carlson (1982) defined a DSS as "an interactive computer-based support system that helps decision makers utilize data and models to solve unstructured problems." A DSS integrates the following interactive subsystems: (1) model base management subsystem; (2) database management subsystem; and (3) dialog generation and management subsystem. As illustrated in Fig. 1, MODSIM embodies all essential components of a decision support system. The graphical user interface (GUI) connects MODSIM with the various database management components and an efficient network flow optimization model. The objective function and constraints of the network flow optimization model are automatically constructed through the GUI without requiring any background in optimization or computer programming by the user. Optimization of the objective function essentially provides an

Fig. 1. K-MODSIM as a Decision Support System

Dialog Subsystem

Database Subsystem

Models Subsystem

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efficient means of achieving system targets and guidecurves according to desired priorities, while assuring that water is allocated according to physical, hydrological, and institutional/administrative aspects of river basin management.

II. MODSIM 8.0 FEATURES AND APPLICATIONS A. MODSIM 8.0 Software Development The MODSIM 8.0 DSS operates under Microsoft Windows ME, 98, NT, 2000, and XP, although a LINUX version is also available. The Windows version is comprised entirely of native, object-oriented code written in Microsoft Visual C++.NET. The graphical user interface for MODSIM is developed in Visual Basic.NET, and includes both native code and code requiring a developer license, but allowing free distribution of runtime applications without imposing licensing requirements or any costs to the user. The Visual Studio .NET IDE facilitates development and management of large-scale applications, including improved reliability, scalability, security, and performance. The .NET Framework provides a new architecture that overcomes restrictions of COM-based technology, while still allowing COM interoperability for development of managed code that uses COM components. The .NET Framework provides a single API instead of having to consume a number of API’s such as Win32, ole32.dll, etc. Managed Code running under the Common Language Runtime (CLR) of .NET provides platform independence, language interoperability, and performance improvement. One of the greatest advantages of .NET is the ability of the user to customize MODSIM for any specialized operating rules, input data, output reports, and access to external models running concurrently with MODSIM, all without having to modify the original MODSIM source code. Customized code can be developed in any of the several .NET languages that are freely provided with .NET Framework. All PUBLIC variables and object classes in MODSIM are directly accessible to the custom code, and the .NET CLR produces executable code as opposed to other applications requiring scripts to be prepared in an interpreted language such as PERL or JAVASCRIPT with poorer runtime performance. MODSIM is distributed as freeware on the Internet at: http://modsim.engr.colostate.edu. The MODSIM installer provides the option of installing the Microsoft .NET Framework if it is not currently installed on the target machine. Also downloadable at this site are K-MODSIM user manuals, documentation, example applications, and sample data sets. B. MODSIM 8.0 Capabilities MODSIM 8.0 includes many unique features and capabilities not found in other river basin management models:

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Microsoft .NET Framework. MODSIM is implemented on desktop computers operating under Microsoft Windows ME, 98, NT, 2000 or XP, and is developed under the new Microsoft .NET Framework. Freeware. Since all components of MODSIM are developed entirely from native Visual C++.NET code, there is no need for MODSIM users to purchase expensive licenses for proprietary software. Graphical User Interface. MODSIM employs a powerful, interactive graphical user interface (GUI) for creating, locating and connecting river basin network components, as well as a spreadsheet-style data editing capability emulating an object-oriented database management system. Time series data sets are imported into MODSIM from external database management systems. Network Flow Optimization. The basic solver in MODSIM is a state-of-the-art network flow optimization algorithm that is more than an order of magnitude faster than solvers currently in use in other river basin modeling packages and capable of modeling extremely large-scale networks. Data-driven Model. MODSIM maintains complete reliance on user input data and specifications for describing system features, operational requirements, and priorities, which are separated from the network modeling algorithmic structure; no a priori defined operating policies or priorities are built-in to MODSIM. Long-term to Real-time Operations. MODSIM is applicable to long term planning (monthly), medium term management (weekly), and short term operations (daily) in river systems. Complex River Basin Configurations. MODSIM allows simulation of a wide variety of river basin configurations and operating conditions without requiring specification of complex IF-THEN rules governing allocation of flows and storage. Complex network topologies can be constructed, including looped and bifurcating flow paths. Network topology is graphically created by simple point and click actions on the GUI palette. In addition, georeferenced network structures can be loaded into MODSIM from a geographic information system (GIS). Reservoir Operations and Hydropower Generation. Reservoir balancing routines are included, allowing division of reservoir storage into several operational zones for controlling the spatial distribution of available storage in a river basin. Hydropower generation capacity and energy production is based on power plant efficiencies varying with flow, head, and load factor. On-peak vs. off-peak and firm vs. secondary energy production calculations are performed, including consideration of tailwater effects and head-dependent hydraulic capacity restrictions on reservoir discharge. Conditional Operating Rules. Operating rules on reservoir regulation and demand allocation can be conditioned on user defined system hydrologic state information for the

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current period, including development of shortage rules for equitably distributing water demand shortages over a basin during low-flow or drought conditions. Watch Logic. A highly functional watch logic calculator offering several algebraic and logical operators is included in the MODSIM GUI that allows user-specified water allocation rules to be based on flow and storage conditions anywhere in the river basin network. Conjunctive Use. MODSIM includes modeling capabilities for conjunctive use of surface water and groundwater and simulation of stream-aquifer interactions. A stream-aquifer model based on the USGS sdf approach is included, as well possible linkage with external groundwater models. A GIS tool called MAPSIM is included with MODSIM that provides processing of spatially distributed stream-aquifer response functions obtained from numerical 3-D groundwater models such as MODFLOW. Water Rights and Storage Contracts. MODSIM is capable of directly incorporating institutional structures governing water allocation under direct flow or natural streamflow rights and seasonal storage rights and contracts, including provisions for allocating water according to specified priorities based on current river basin conditions. Other administrative mechanisms that can be modeled in MODSIM include rent pools, water banking, flow augmentation plans, and exchanges that allow flexible system operations while maintaining water rights and contract legality. Customized MODSIM. Users can prepare customized code in the Visual Basic.NET or C#.NET languages that are compiled with MODSIM through the Microsoft .NET Framework. Users are provided access to all key variables and object classes in MODSIM, thereby allowing customization for any complex river basin operational and modeling constructs without the need for reprogramming and recompiling the MODSIM source code. Custom code can be developed for defining complex operating rules and policies, executing external modules such as water quality models, input of specialized data sets for particular applications, preparing customized model output and reports, and linking MODSIM with database management systems to provide access to timely data and forecast information for real-time river basin management. Streamflow Routing. MODSIM includes Muskingum or user-specified time-lagged hydrologic streamflow routing capabilities for daily simulation. In addition, an innovative backrouting procedure is available which looks ahead to future time periods in order maintain legal water allocation under appropriative water rights. Monte Carlo Analysis. MODSIM allows simulation of synthetic or stochastically generated inflow/demand sequences for use in Monte Carlo analysis for developing flow-duration curves and exceedance probability estimates for key variables. Graphical Plots. MODSIM produces graphical plots of important model time series variables reflecting system performance, as well as tabulated results showing storage levels, releases, inflows, energy generation, power capacity, system losses and spills,

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water deliveries, shortages, instream flow requirements, and flows in any reach of the system. Plotting packages are available for comparative evaluation of operational plans and scenarios, including display of flow-duration curves and statistical analysis. C. Successful Applications of MODSIM The MODSIM model was originally developed at Colorado State University (CSU) in the late-1970's (Shafer and Labadie, 1978). Since then, MODSIM has undergone dramatic improvements and updates, and has enjoyed widespread use by numerous governmental and private organizations for simulating complex river system operations in the U.S. and throughout the world. Various versions of MODSIM have been successfully applied to a number of complex river basin systems: Rio Grande River Basin (Graham, et al., 1986) sponsor: U.S. Forest Service scope : Entire Rio Grande River Basin in Colorado, New Mexico, and Texas monthly time step: stream-aquifer interactions not included since focus is on

incremental surface water inflows to the basin objective: determination of how additional flows made available through planned

silvacultural activities in the Rio Grande National Forest would be allocated downstream with consideration of complex in-state water right decrees and interstate compact agreements; includes analysis of impact of possible future storage facilities

Upper Pampanga River Basin, The Philippines (Faux, et al., 1986) sponsors: National Science Foundation - International Programs and National

Irrigation Administration, Manilla, The Philippines scope: Upper Pampanga River Basin covering 6700 km2 in Central Luzon;

considered the country’s most important rice producing region monthly time step: surface water modeling only objective: improve operational efficiency of the Upper Pampanga River Integrated

Irrigation System (UPRIIS), balance hydroelectric generation and irrigation supply, and identify bottlenecks in the water distribution network for possible expansion of canal capacity

Colorado-Big Thompson River System, Northern Colorado (Law and Brown, 1989) sponsor: Northern Colorado Water Conservancy District scope: both the West Slope and East Slope components of the Colorado-Big

Thompson (C-BT) project of Northern Colorado monthly time step: surface water modeling only objective: fully integrated operations of the C-BT/Windy Gap system and portions of

the Cache La Poudre River Basin; predict the yield of a proposed reservoir on the Cache La Poudre River and investigate various management options

Lower Nile River Basin, Egypt (El-Beshri and Labadie, 1994) sponsors: Egyptian Ministry of Public Works and Water Resources and USAID scope: Lower Nile River, including the Nile Delta monthly time step: both surface water and groundwater modeling

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objective: investigate the conjunctive use of surface water and groundwater in the Lower Nile Basin in order to reduce pressure on the High Aswan Dam for meeting current and future water supply requirements.

San Joaquin River, Central Valley, California (Leu, 2001) sponsor: U.S. Bureau of Reclamation, Mid-Pacific Region, Sacramento, California scope: San Joaquin River Basin (1,638 mi2), California monthly time step objective: investigate the use of economic-based strategies such as increased water

prices, tiered water pricing, changes in San Joaquin River environmental flows, and changes in reservoir operations to improve water management

Gunnison River Basin, Colorado (Weiss, et al., 1997) sponsor: Colorado Department of Water Resources scope: Gunnison River Basin, tributary to the Colorado River monthly time step: surface water modeling only objective: environmental impact evaluation of proposed AB Lateral Hydropower

Facility, Montrose Colorado; fully integrated modeling of entire river basin including consideration of over 1000 water rights and complex exchange agreements

South Platte River Basin, Colorado (Fredericks, et al., 1998) sponsor: Colorado Water Resources Research Institute scope: Lower South Platte River Basin, Colorado monthly and daily time steps: conjunctive use of groundwater and surface water;

integration of MODSIM and MODFLOW groundwater model; integration of MODSIM and GIS

objective: analysis of groundwater augmentation plans to replace depletions that would otherwise accrue to the South Platte River as a result of well pumping used to meet irrigation demands

Arkansas River Basin (Kastner, 2001) sponsor: Office of the Colorado State Engineer scope: Lower Arkansas River Basin below Pueblo Dam, Colorado daily time step objective: evaluation of the Winter Water Storage Program for allocating and storing

winter season water supplies in the Arkansas River basin among water users who formerly used these waters for direct flow winter irrigation; the stored waters are later used more effectively and efficiently during the following irrigation season.

Imperial Irrigation District, California (CH2MHill, 2001) sponsor: Imperial Irrigation District (IID), El Centro, California scope: District irrigated area of over 460,000 acres; average annual flow of 3,000,000

acre-ft diverted by IID at the All American Canal

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monthly and daily time step: MODSIM network models 230 miles of main canal, 1,440 miles of secondary canals and laterals, and 5,590 delivery gates; applied to both the water distribution system and the irrigation drainage system; MODSIM network composed of over 10,000 nodes and links; integration of water quality models into MODSIM to assess impacts of water conservation programs on drainage water quality, which impacts the nearby Salton Sea

objective: apply MODSIM to assessing both the water quantity and quality impacts of voluntary conservation programs that could provide transfer of up to 300,000 acre-ft of water to metropolitan areas of Southern California

Upper Snake River Basin, Idaho (Larson and Spinazola, 2000; Miller, et al., 2003) sponsor: U.S. Bureau of Reclamation, Pacific Northwest Region scope: Upper Snake River Basin, Idaho monthly time step: conjunctive use of groundwater and surface; integration of

MODSIM and MODFLOW groundwater model objective: quantification of impacts to irrigation water supply, river and reservoir

recreation, resident fish and wildlife, and other local water uses from various proposed storage rental and reallocation scenarios for satisfying instream flow requirements for endangered species.

Piracicaba River Basin, Brazil (de Azevedo, et al., 2000) sponsor: National Council of Science and Technology, Brazil scope: Piracicaba River Basin (12,400 km2), State of Sao Paulo, Brazil monthly time step: integration of water quantity and quality modeling objective: joint application of MODSIM and the QUAL2E-UNCAS stream water

quality model for evaluation of strategic planning alternatives for meeting transbasin diversion requirements for the city of Sao Paulo, intrabasin water supply needs, and acceptable water quality according to various reliability criteria

Deschutes River Basin, Oregon (La Marche, 2001) sponsors: Oregon Water Resources Department and U.S. Bureau of Reclamation scope: Upper and Middle Deschutes Basin and Crooked River Basin; includes two

major reservoirs: Wickiup (200,000 ac-ft) and Crane Prairie (55,800); beneficial uses primarily irrigation (100,000 acres of irrigated lands), recreation, fish and wildlife maintenance, and flood control

monthly time step objective: optimal allocation of water in the Deschutes River Basin to satisfy both

irrigation demands and instream flow requirements Lower Arkansas River Basin, Colorado (Dai and Labadie, 2001) sponsor: Colorado Water Resources Research Institute scope: Lower Arkansas River Basin below Pueblo Dam, Colorado monthly time step: conjunctive use of groundwater and surface water; integration of

water quality and water quantity modeling with MODSIM

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objective: Determine opportunities for improving water quality in the Lower Arkansas River Basin through conjunctive use of groundwater and surface water

Klamath River Basin, Oregon and California (Campbell, et al., 2001) sponsor: U.S. Geological Survey, Biological Resource Division scope: Klamath River Basin from Keno, Oregon to Seiad Valley, California monthly and daily time step: integration of MODSIM with water quality HEC-5Q

model objective: integrated application of MODSIM and HEC-5Q to explore potential for

changing system operations to improve summer/fall water quality conditions to benefit declining anadromous fish populations

Little Butte and Bear Creek River Basins, Oregon (Stillwater, 2003) sponsor: U.S. Bureau of Reclamation, Pacific Northwest Region scope: Little Butte and Bear Creek Rivers, tributary to the Rogue River, Oregon

(includes over 37,000 acres of irrigated lands); includes transbasin diversions from the Klamath River Basin

monthly time step: separates USBR project water from natural flow rights and includes numerous storage accounts in several reservoirs in the basins.

objective: apply MODSIM for the Little Butte/Bear Creeks Management Project Steering Committee to demonstrate the effects of saved water and alternative and supplemental water supplies. The irrigation districts and other local irrigators, the State water master, and technical specialists from Federal and State natural resource agencies, provided direction and input for MODSIM network development.

Payette River Basin, Idaho (Stillwater, 2004a) sponsor: U.S. Bureau of Reclamation, Pacific Northwest Region scope: physical and operational characteristics of reservoirs, river reaches, and

diversions of the Payette River system; reservoirs modeled include Cascade Reservoir, Deadwood Reservoir, Payette Lake and the Upper Lakes, and Black Canyon Reservoir.

monthly time step: historical streamflow record extending from 1928 to present objective: determine the impacts of streamflow augmentation for endangered species

on existing irrigation water supplies Tualatin River Basin, Oregon (Stillwater, 2004b) sponsor: U.S. Bureau of Reclamation, Pacific Northwest Region scope: Tualatin River Basin (700 mi2 drainage); includes U.S. Bureau of Reclamation

projects Barney Reservoir (20,000 ac-ft) and Hagg Reservoir (53,640 ac-ft), providing water for irrigation, municipal supplies, stream quality, flood protection and recreational benefits.

monthly time step: includes conjunctive use of groundwater and surface water objective: demonstrate the effects of Reclamation’s current operations in the Tualatin

Basin

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In addition to these applications, MODSIM is serving as a valuable water supply planning tool for several municipalities, including the City of Colorado Springs, Colorado (contact: Brett Gracely, Water Resources Planning Supervisor, Colorado Springs Utilities), the City of Ft. Collins, Colorado (contact: Dennis Bode, Water Resources Manager, City of Ft. Collins), and the City of Greeley, Colorado (contact: Todd Williams, Water Resources Manager, City of Greeley).

III. River Basin Network Development in MODSIM 8.0 A. Network Flow Approach to River Basin Modeling The basic principle underlying MODSIM is that most physical water resource systems can be simulated as capacitated flow networks. The term capacitated refers to imposition of strict upper and lower bounds on all flows in the network. Components of the system are represented as a network of nodes, both storage (i.e., reservoirs, groundwater basins, and storage right accounts) and non-storage (i.e., river confluences, diversion points, and demand locations), and links or arcs (i.e., canals, pipelines, natural river reaches, and decreed water rights) connecting the nodes (Fig. 2). Although MODSIM is primarily a simulation model, the network flow optimization provides an efficient means of assuring allocation of flows in a river basin in accordance with specified water rights and other priority rankings.

Fig. 2. Nodes and links in K-MODSIM

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A network formulation of a river basin system provides a physical picture revealing the morphology of the system that is readily recognizable. In effect, the graphical network links are the model decision variables. Network optimization techniques are specialized algorithms that perform integer-based calculations on linear networks that are considerably more efficient than real number computations and matrix operations employed in standard linear programming codes based on extensions of the revised simplex method. Integer-based calculations are not a disadvantage since appropriate scaling of link flows can produce solutions for any desired order of accuracy. The high efficiency of network flow optimization algorithms allows rapid solution of large-scale networks comprising thousands of nodes and links on desktop computers. This also makes it feasible to perform several iterative solutions so as to consider certain nonlinear or dynamic system features. Important assumptions associated with MODSIM are listed as follows: All storage nodes and linkages must be bounded from below and above (i.e.,

minimum and maximum storage and flows must be given, with the latter allowed to vary over time,

Each linkage must be unidirectional with respect to positive flow; possible flow reversals can be modeled by assigning an additional reverse direction link between two nodes;

All inflows, demands, system gains and losses must accumulate at nodes; increasing the density of nodes in the network thereby increases simulation accuracy, but also increases computer time and data requirements;

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Each reservoir is designated as a spill node for losses from the system proper. Spills from the system are the most expensive type of water transfer, such that the model always seeks to minimize unnecessary spills. Spills may be retained in the network by specification of an additional release link from a reservoir which can be labeled as a high cost link

B. MODSIM 8.0 GUI for Network Creation and Editing The graphical user interface (GUI) for MODSIM 8.0 as shown in Fig. 2 provides spatially-referenced database capabilities allowing users to create and link river basin network objects on the display, and then populate data for that object by right-mouse click to activate the object and open a database form associated with that object. Lengthy time series data for streamflows, demands, etc., can be loaded by copying data from EXCEL or database management software to the Windows clipboard, and then pasting the data into the appropriate Node Properties form. Time series data can also be loaded automatically through development of custom code, as discussed subsequently. All MODSIM input data are stored in a command-oriented ASCII text file *.xy where each line of input begins with a command that the input parsing code associates with a model construct, with data values relevant to the modeled feature following the command. The main Menu Bar for MODSIM is located at the top of the Network Editor Window in the interface, along with a Toolbar below the Menu Bar with single click access to several menu items. The menu items are used to load and save a MODSIM network, import and export data, select English or metric units, search for specific nodes and links,

Fig. 2. Graphical user interface for K-MODSIM

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run the model, select and display graphs; create, edit and generate tabular reports; access various utilities, print out the network, and more. The interface contains icons in the Node Palette Window for creating various types of nodes in the network by simply dragging them into the Network Editor Window, or left-button mouse clicking on the icon and then clicking on the desired location in the Network Editor Window. Links or arcs are created in the Network Editor Window by moving the cursor onto the origin node until a pointing hand icon appears, holding down the left-mouse button, and then dragging the pointer to the desired ending node, which also sets the flow direction for that link. The Network Overview Window is useful for large networks where the display window of any size can be panned over any portion of the network. Clicking on any node object in the Network Editor Window makes that object active, which is indicated by the node being highlighted in a yellow box. Once a node is located in the Network Editor Window, it can be easily moved by left-button mouse click and hold, and then dragging it to the desired location. All links connected to that node will be moved along with the node. Any actions within the Network Editor Window can be undone by the user by selecting Edit >Undo in the Main Menu Bar. Right-button mouse click on any node in the Network Editor Window displays a context menu with several items. Cut allows the node to be pasted at another location, whereas Delete removes the node without paste retrieval. Copy To allows all of the attributes of the selected node to be copied to another user-specified node, whereas Copy From allows all properties of another node to be copied to the current selected node. Hide Label is useful for removing excess notation in the Network Editor Window for nodes where labels are not important for display. Selecting Properties opens the database form for that node, allowing the user to enter and edit all of the data and attributes associated with that object. The Properties Form can also be directly activated by double clicking on the node object. After a MODSIM run is executed, right-button mouse click on a node again opens the context menu, but with an added item: Graph, which allows rapid display of output results. Clicking on any link object in the Network Editor Window makes that object active, which is indicated by display of yellow square-shaped break point markers along the link, as well as yellow diamond-shaped markers at the link beginning and end points. Break points can be moved by simply clicking on them and dragging, thereby allowing users to change the shape of the link. By default, when a link is first created a single break point is inserted in the center of the link, but additional break points can be inserted by the user. Similar to nodes, right-button mouse click on links also opens a context menu with items Delete, Copy To, Copy From, Insert Point, Remove Segment, Convert to MultiLink, and Properties. Similar to the node context menu, once a MODSIM run is completed, the Graph item is added to the link context menu. Insert Point allows users to introduce additional break points in the link, thereby changing its shape as connected line segments. Remove Segment performs the opposite operation by removing any segment. Convert to MultiLink allows the user

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to specify any desired number of separate links connecting the same two nodes, and Properties opens the database form, allowing the user to enter and edit all data and attributes associated with that link object. Once a link object is converted to a MultiLink, it is displayed as a double arrow in the Network Editor Window. In addition, for simulations using a daily time step, any link can be specified as a channel routing link in the Link Properties Form, which changes the color of the link to orange. As with nodes, the Link Properties Form can also be directly activated by double clicking on the link object. Similar to the Node Properties Form, the Link Properties Form is tabbed for each data category, with spreadsheet-style data entry for tabular data. Under the Edit Pull-Down Menu, a number of useful operations on network objects are available, such as Undo, Cut, Copy, Paste, Delete, Resize Nodes and Links, Node and Link Search, and various Select options. Node Search and Link Search is particularly useful for large networks, where a particular node and link can be selected from a scrollable list or directly entered by name or number, where each network object created in MODSIM is automatically assigned a unique number. The network display then shifts to the region surrounding the selected node or link, which is highlighted as an active object. Select Nodes and Select Link is useful for output of results for only those objects of interest in the simulation. The View Menu Option on the Main Toolbar provides a number of additional features, including Zoom operations, Messages for display of information concerning the status of the MODSIM run, and Snap to Grid, which when activated, snaps created or moved objects in the Network Editor Window to a finite grid for ease of alignment of objects. Unchecking Snap to Grid allows objects to be moved to any location in the Window.

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Once the system network has been created and the database populated, the MODSIM model can be executed from the interface under MODSIM>Run MODSIM. A number of useful Extensions are included with MODSIM, including the Storage Rights Extension, Back-Routing Extension, Water Rights Extension, and Last-Fill Extension. MODSIM>Custom Runs invokes the Custom Code Editor for creating customized versions of MODSIM developed from user-supplied code written in VB.NET or C#.NET. Customized versions of MODSIM can be compiled and executed from the interface. A convenient template is provided in the Custom Code Editor for guiding users in the preparation of customized code. The customized code can interface with MODSIM at any desired strategic locations, including data input, execution at the beginning of any time step, processing at intermediate iterations, and model output. Users are provided direct access to all of public variables, parameters, and object classes in MODSIM for development of knowledge-based operating rules, linkage with on-line database management systems, customized output reports, and color-coded graphical displays. Output Control provides an extensive variety of graphical and text output options are available for any combinations of network objects and output data types. In the production of output results, the user can specify the number of time steps held in memory. Retaining output results over several time steps in main memory generally results in faster execution speed, but also requires larger

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C. Network Flow Optimization Links and nodes in MODSIM are not confined to representing physical and hydrologic features of a river basin system, but are also used to symbolize artificial and conceptual elements for modeling complex administrative and legal mechanisms governing water allocation. In addition to the links and nodes defined by users, several artificial nodes and links are automatically created by MODSIM, as shown in Fig. 3. These artificial nodes and links are essential to insuring mass balance is satisfied throughout the entire network. It should be noted that MODSIM users are only responsible for defining the physical flow network. All artificial nodes and links are added automatically by the model. MODSIM simulates water allocation mechanisms in a river basin through sequential solution of the following network flow optimization problem for each time period t = 1,...,T :

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minimize k kk A

c q∈∑ (1)

subject to: ( ) for all nodes

i i

k itk O I

q q b i N∈ ∈

− = ∈∑ ∑ q (2)

( ) ( ) for all links kt k ktl q u k A≤ ≤ ∈q q (3)

where A is the set of all arcs or links in the network; N is the set of all nodes; Oi is the set of all links originating at node i (i.e., outflow links); Ii is the set of all links terminating at node i (i.e., inflow links); bit is the (positive) gain or (negative) loss at node i at time t; qk is the flow rate in link k ; ck are costs, weighting factors, or water right priorities per unit flow rate in link k; and lkt and ukt are specified lower and upper bounds, respectively, on flow in link k at time t. Note that parameters bkt , lkt, ukt are defined as functions of the flow vector q in the network. These nonlinearities are due to flow dependent calculation of evaporation (based on flow in the carryover storage artificial arcs shown in Fig. 3), groundwater return flows, channel losses, and instream flow requirements, and are primarily associated with the artificial arcs. A successive approximations solution procedure is adopted whereby an initial set of flows q are

Fig. 3. Illustration of MODSIM network structure with artificial nodes and links

Active Storage Zone n

Stream Depletion

3

4

GW

SP

M

D S

Initial Storage plus Inflow

Local Inflow

Total Inflow plus Initial Storage

Evaporation

Flood Storage

Active Storage Zone 1

…

Return Flow

Spill

Mass Balance Demand

Storage

Pumping

Infiltration

Channel Loss Seepage

1

2

31 2

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assumed, resulting in initial estimates of the flow-dependent parameters bit, lkt, ukt . Eqs. 1-3 are then solved with the highly efficient Lagrangian relaxation algorithm RELAX-IV (Bertsekas and Tseng, 1994), which is up to two orders of magnitude faster than the revised simplex method of linear programming. The flows q produced from this solution then serve to update estimates of parameters bit, lkt, ukt , and the network flow optimization repeats until convergence. Optimization is primarily conducted as a means of accurately simulating the allocation of water resources in accordance with operational priorities based on system objectives, operational experience, water rights, and other ranking mechanisms, which may include economic factors. The network topology and object characteristics are defined by the sets N, A, Ii, and Oi and arc parameters [lkt , ukt , ck] for each arc or link k, for each period t. Since solution of Eqs. 1-3 is executed period by period, rather than as a fully dynamic optimization, flows in the carryover storage arcs (Fig. 3) become initial storage levels for the next period optimization.

IV. GENERAL SETTINGS FOR MODSIM 8.0 SIMULATION A. Accuracy and Units Network Settings under MODSIM on the main menu bar allows users to specify a number of options. The desired accuracy for data input and flow results can be either integer accuracy or 2-place decimal accuracy. Users can select either English or metric units, with the units used for storage, flow, head, net evaporation rates and hydropower specified.

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B. Time Steps and Scale MODSIM>Network Settings allows monthly, weekly, or daily time steps specified.

Users can enter Start Date and End Date for time series data entered into MODSIM, which must be the same for all data including inflows, demands, reservoir storage targets, etc.

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However, a particular MODSIM simulation run can start and end at any intermediate dates, as shown under the General Tab under MODSIM>Network Settings. This is useful, for example, for critical period studies covering only drought sequences within the entire historical hydrologic data set. Notice that MODSIM provides a convenient calendar for specifying these dates.

V. HYDROLOGIC AND ADMINISTRATIVE COMPONENTS

A. Unregulated Inflows Native or unregulated hydrologic inflows are input to MODSIM from measured flow data, watershed runoff models, forecasts, drought scenarios, or stochastic generation of streamflows. Unlike other river basin models, the inflows supplied to MODSIM are unregulated, incremental, or localized inflow gains to a river reach. River basin models such as TAMUWRAP (Wurbs and Walls, 1989), WIRSOS (State of Wyoming, 1992) and STATEMOD (State of Colorado, 1999) require the user to develop virgin or undeveloped flow conditions in the basin prior to application of the model, which are often difficult to synthesize. Larson (2003) applied the SAMS software package (Salas, et al., 2002) for stochastic generation of streamflow gains to the Snake River basin for a MODSIM application, and Labadie, et al. (2005) applied SAMS to stochastic generation of streamflows for application of MODSIM to the Geum River Basin, Korea. Inflows are assigned as right-hand-side constants in Eq. 2 for both storage and nonstorage nodes, with initial storage for the current period added to the inflows for storage nodes. for storage node it it itb S I i= +

for nonstorage node it itb I i=

where itI is local inflow to node i during period t and itS is storage in reservoir node i at

the start of period t.

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Double-clicking any NonStorage Node in the Network Editor Window displays the NonStorage Properties Form, allowing unregulated inflows to be entered, as shown in Fig. 4. Data can be entered manually, or right-button mouse click on the heading of the inflow data column heading produces a context menu allowing data to be copied from a spreadsheet, database, or ASCII text file into the Windows Clipboard to be pasted into this form. Alternatively, lengthy database files may be automatically loaded into MODSIM using custom code written in one of the .NET languages. As seen in Fig. 5, an option available in the NonStorage Node Properties Form is to generate a graphical plot of the time-series data entered in the Form. This is useful as a visual means of scanning the data for obvious outliers and inconsistent data. Holding down the left-button mouse and dragging a box clockwise over the graph will zoom in to the selected portion. Dragging a box counterclockwise through the origin will return the graph to its original resolution.

B. Reservoir Operations Reservoir Storage Zones. Reservoir maximum volume Si,max, minimum volume Si,min, initial volume Si0, and reservoir priority number are entered into the General Tab of the Reservoir Node Properties Form, as seen in Fig. 6, where the artificial carryover storage link cost is calculated as:

Fig. 4. Properties form for inflow data at nonstorage node.

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( )50000 10i ic OPRP= − − ⋅ (4)

where OPRPi is a priority ranking from 1 to 5000, with lower numbers indicating a higher ranking, resulting in a negative cost. Notice that minimization of negative costs in Eq. 1 is equivalent to maximizing flows to the higher ranked water uses. Rather than their absolute values, it is the relative order or ranking of the negative costs that determines how MODSIM allocates network flows. Fig. 7 shows the artificial carryover storage links for each storage zone originating at each reservoir and accumulating at artificial carryover storage node S. The link or arc parameters are displayed in Fig. 7, where (negative) costs for the storage zone arcs are incremented from the negative cost associated with maintaining the active storage target in the reservoir.

Fig. 5 Graphical display of inflow data at nonstorage node.

24

The bounds on the zone carryover storage links represent the incremental storage in each zone as defined by the user in the MODSIM GUI, where Si,min is minimum allowable storage in storage node i, Si,max is maximum storage, Tit is ideal target storage level for storage node i for current period t, αi is the fraction of either maximum capacity or target storage defining zone layer i, and ∆ci is the corresponding incremental (negative

Fig. 6. Reservoir capacity, initial storage and priority

Si,max

Si,min

Si0

OPRPi

Fig. 7. Link parameters for zone carryover storage arcs.

S

2 10 1α α< < <

1 20 c c>∆ >∆

,min 2 2, ,i i iS T c cα ⋅ +∆

( )1 2 10, ,i iT c cα α − ⋅ +∆

( )10, 1 ,i iT cα − ⋅

,max0, , 0i iS T −

,miniS

,maxiS

iT1 i

Tα ⋅2 i

Tα ⋅

Flood Pool

Zone 1 Zone 2

Zone 3

Active storage

Dead storage

i

25

cost) for zone i, which are entered into the Reservoir Node Properties Form as shown in Fig. 8. To achieve balanced storage among several reservoirs in a basin, each reservoir should have the same OPRPi priority and similar incremental costs ∆ci.

Flood Operations. Although MODSIM is not designed for flood operations studies, flood pool zones are defined above the target storage levels in reservoirs, as seen in Fig.7. These target storage levels can vary seasonally for securing increased space during the flood season. As seen in Fig. 7, the flood carryover storage link is assigned a zero cost in MODSIM. In conjunction with the flood space, the final terminal node in a river basin should be a Network Sink Node, as shown in Fig. 9. The Network Sink Node is assigned a default priority number of 4999, resulting in a small negative cost of -10 on

1100α ⋅

2100α ⋅

3100α ⋅3

c∆

2c∆

1c∆

Tit

Fig. 8. Reservoir target storage levels and reservoir layer properties.

26

the artificial demand link conveying flows out of the basin (Eq. 4). A high demand is assigned to the Network Sink Node, assuring feasible network solutions under high flow conditions. Since the artificial flood zone carryover arcs are assigned a zero cost, any temporary storage in the flood zone is released downstream as soon as sufficient conveyance capacity is available. In addition, as seen in Fig. 3, high cost artificial spill links convey excess flows out of the basin only if available flood space is filled and downstream conveyance channels are at maximum capacity. If it is desired to retain excess spill flows in the basin, parallel, high cost links can be created downstream to represent overbank flooding. Conditional Operating Rules. Target storage levels itT can be input as a time series of ideal storage levels, as shown in Fig. 8, or as a set of rules conditioning target storage settings on user defined hydrologic state information at the current time step. The former approach is often utilized for calibrating MODSIM by specifying target storage levels as measured historical data and then adjusting various parameters in MODSIM to match available stream gage records. The use of hydrologic state information is valuable for management simulation runs of MODSIM after calibration is completed. MODSIM computes hydrologic states by considering current reservoir storage levels and current period inflows to a certain user specified subset of reservoirs in the system that are

Fig. 9. Network sink node located at basin outlet.

27

Dry

B1τm Bi-1,τm Biτm Bn-1,τm

Ave.

Hydrologic State Rtm

Wet

Fig. 10. Definition of Hydrologic States (assuming period t is in calendar month τ).

Tar

get

Sto

rag

e T

i t

indicative of hydrologic conditions in the basin. Several different hydrologic state subset designations may be specified as needed. Associated with each of these states (which may be classified as average, dry, wet, etc.) is a corresponding set of reservoir operating rules with associated ranking priorities. These hydrologic states are computed at the beginning of each period for the user selected reservoir subset through the following analysis:

[ ]m

tm it iti H

R S I∈

= +∑ (5)

,max

m

m ii H

W S∈

= ∑ (6)

where Hm is the set of node numbers of reservoirs in a specified subset defining hydrologic state designation m; t is the current period of operation; Iit is the unregulated inflow to reservoir i for period t ; Sit is the beginning storage in reservoir i, period t; and Si,max is the storage capacity for reservoir i. The ranges for each hydrologic state designation are defined by user input boundary factors i mτβ (i = 1,…,n-1) as fractions of total subsystem storage capacity for seasonal period τ (i.e., calendar month for monthly time steps in the simulation), where 1 1,0 m i m n mτ τ τβ β β −≤ < < < < Boundaries dividing the hydrologic state ranges are then calculated as: (7) where n is the number of hydrologic states in designation m; i mBτ is the upper bound on

hydrologic state i for period τ. As shown in Fig. 10, the n hydrologic state ranges for seasonal period τ are defined as:

1

1,

1,

: 0

:

:

tm m

i m tm i m

n m tm

Dry R B

Medium B R B

Wet B R

τ

τ τ

τ

−

−

≤ ≤

≤ ≤

≤

where period t is assumed to be in calendar month τ and reservoir targets itT are constant with these

hydrologic states. Conditional target storage levels can only vary within a computational cycle (i.e., one year for monthly analysis), although

for 1,..., 1i m i m m

n m m

B W i n

B Wτ τ

τ

β= = −=

28

separate target storage levels can be specified for each hydrologic state. MODSIM also allows differing priorities to be specified for any reservoir node corresponding to hydrologic state conditions as calculated by the above procedure. Notice in Fig. 11 that when Conditional Rules is selected as Run Type in the Network Settings Form, a new Hydrologic State Tab is displayed, which displays a list of the

existing Hydrologic State Tables (Fig. 12), and from which new tables can be created.

Fig. 11. Run Type option: Explicit Targets or Conditional Rules.

Fig. 12. Creation and editing of Hydrologic State Tables.

29

Clicking Create Table or Edit Table displays the Hydrologic Table Form (Fig. 13) which allows designation of the hydrologic state subsystem of reservoirs for conditional rules, as well as the boundary factors defining the range of each hydrologic state. Boundary factors i mτβ may be > 1 since reservoir operating targets are conditioned on

total current storage plus total inflows for reservoirs in the designated hydrologic state subsystem. Since upper bounds i mBτ are calculated by multiplying boundary factors

i mτβ by total storage capacity Wm, without inclusion of inflows, higher boundary factors

account for the volume of unregulated inflows entering the reservoir.

Once hydrologic state tables have been created, the desired Hydrologic Table is easily selected from the drop down list in the form activated by clicking the General Tab in the Reservoir Node Properties Form. As shown in Fig. 14, varying priorities can be specified for each hydrologic state. The conditional reservoir operating targets can then be input for each hydrologic state into the form activated by clicking the Targets Tab, as shown in Fig. 15. Again, these targets are only applied to the seasonal time steps, i.e., the calendar months of each year. However, since it may be preferred that some reservoirs in

H1 Subsystem designation

n

β1τm τ

β2τm β3τm

Fig. 13. Designation of Hydrologic State subsystem and boundary factors.

30

the system are operated with explicit rather than conditional target storage levels, the Varies By Year checkbox allows input of explicit targets for each time period. Inflow Forecasts. It is possible to utilize the hydrologic state option in for incorporation of seasonal inflow forecasts in specification of reservoir target operating rules. This is accomplished by adding a dummy reservoir with zero capacity to the network, but not directly connecting it to the network proper. As can be seen in Fig. 16, a dummy reservoir HeiseFor is created for the Upper Snake River Basin in Wyoming, USA which is disconnected from the rest of the network. Seasonal runoff forecasts are entered into the form activated by the Runoff Forecast tab. Seasonal forecasts are associated with the dummy reservoir, but the runoff forecasts do not provide actual inflow to this node. This dummy reservoir may be included in the hydrologic state subsystem designation of reservoirs and used for developing conditional reservoir operating targets based on anticipated future inflows entering the system. This allows operating rules to be based on forecasts of runoff conditions, such as information obtained from snow pack surveys in the watershed. Data entered into the Runoff Forecast form represents seasonal forecast information that would be available at each time period of operation. If desired, Runoff Forecasts can be entered into actual reservoirs that are directly connected to the network. Again, these forecasts do not provide actual inflows to the reservoir, but can be used to include that reservoir in a hydrologic state designation which, in this case, would also include the current storage in the reservoir. Because these Runoff Forecast data are not provided as actual inflows to the system, they can be represented as subjective indices, such as 1: representing low flows, 3 representing average flows, 5: representing high flows, etc.

Fig. 14. Selecting Hydrologic Table Name for conditional reservoir operating targets.

31

Fig. 15. Specification of reservoir target storage levels conditioned on each Hydrologic State.

32

Evaporation and Seepage Losses: Users enter net evaporation rates in the Reservoir Node Properties form, which are defined as evaporation rates minus rainfall rates. As seen in Fig. 17, negative entries signify that rainfall rates exceed evaporation rates for that time period. MODSIM accepts a variable number of elevation-area-storage-hydraulic outlet capacity data points for any reservoir, as shown in Fig. 18, which are interpolated to calculate reservoir surface area corresponding to any current volume in the reservoir. Evaporation loss is calculated in MODSIM as a function of average surface area in a reservoir over the current period. , 10.5 ( ) ( )it it i it i i tEV ev A S A S + = ⋅ ⋅ + (8)

where evit is net evaporation rate (i.e., evaporation rate less rainfall rate) for reservoir i (e.g., feet per month) for the current period t; Ai(Sit) is the interpolated calculation of surface area from the elevation-area-storage-hydraulic outlet capacity table for reservoir i, and Sit is storage at the beginning of the current period. Since average surface area in a reservoir is unknown until calculations are completed for the current period, an iterative process is required for accurate calculation of net evaporation loss whereby net evaporation loss itEV is initially calculated using surface area based on the beginning

Fig. 16. Incorporation of runoff forecast information in Reservoir Node Properties form.

33

Ft. (Evaporation – Rainfall)

Fig. 17. Net evaporation rates entered into the Reservoir Node Properties form.

Fig. 18. Area-Capacity-Elevation-Hydraulic Capacity table.

34

storage volume in the reservoir. With calculation of the ending storage volume for that period , 1i tS + (or beginning storage volume for the next period), an updated estimate of net evaporation loss itEV is calculated based on ( ), 1i i tA S + and a new network solution is obtained based on the updated evaporation loss term. This procedure is repeated until the net evaporation loss calculations converge to stationary values within a desired order of accuracy. Seepage losses from reservoirs are calculated in an iterative process similar to that of evaporation losses, except that seepage is assumed to be a function of average volume in the reservoir over the current period and the seepage loss rate is assumed to be constant:

, 10.5it i it i tSL sl S S + = ⋅ ⋅ + (9)

where sli is the seepage loss fraction for reservoir i and SLit is the total volume of seepage loss during period t. In contrast with evaporation losses, a portion of reservoir seepage may be designated as contributing to downstream return flows, as discussed subsequently. The seepage loss fraction is entered in the form activated by the Groundwater Seepage tab in the Reservoir Node Properties form, as seen in Fig. 19.

C. Hydropower High-Head Power Plants. MODSIM computes both power capacity and energy generation in high-head power plants using the basic power equation: ( ) ,max,it it it i it it iP K Q H e Q H P= ⋅ ⋅ ⋅ ≤ (10)

where Pit is mean power output during period t (KW); Qit is turbine release

(volume/time period); itH is mean effective head for time period t; ( ),i it ite Q H is overall

plant efficiency, which can be entered as a table of values as a function of discrete release rates Q and heads H ; K is conversion constant based on selected units; and ,maxiP is the

maximum capacity of the powerplant. The Power tab in the Reservoir Node Properties form activates the form for input of power plant information, as seen in Fig. 20.

Fig. 19. Input of reservoir seepage rate in the Reservoir Node Properties form.

35

Mean effective head is calculated as:

( ), 10.5 ( ) (it i it i i tH H S H S += ⋅ + (11)

and effective head ( )i itH S is defined as: (water surface elevation) – (power plant

elevation) – (tailwater head loss), where water surface elevation is interpolated from the Area-Capacity-Elevation-Hydraulic Capacity table as in Fig. 18 for the current storage level in reservoir i, and tailwater head loss is calculated as:

tailwater head loss = max (0., tailwater elevation – power plant elevation) (12)

Users can enter tables relating tailwater elevation to discharge for calculation of tailwater head loss, as shown in Fig. 20.

102000

0

Fig. 20. Power plant information

36

On-Peak Energy Generation. Energy generation (MWH) is calculated as:

it it itE P T= ⋅∆ (13)

where itT∆ is total hours of on-peak generation in time period t, as entered into the

Generating Hours form as shown in Fig. 21. MODSIM assumes that hydropower plants are operated primarily for on-peak energy generation, where for monthly time periods, the monthly plant factor is defined as:

,max

it

i it

Eplant factor

P T=

⋅∆ (14)

The Power form under the Reservoir Node Properties form (Fig. 20) includes a check box: Peak Generation Only. Clicking this checkbox signifies that discharges from the reservoir only occur during the on-peak hours of generation. In effect, discharge rate during the on-peak hours only is now calculated as:

p titit

it

HQ Q

T= ⋅

∆ (15)

where itQ is the average discharge rate through the power plant in reservoir i during

period t; tH is the total hours in period t; and pitQ is the discharge rate occurring during

the on-peak hours. This increases power capacity during the on-peak hours according to the following formula:

( ),p pit it i itit itP K Q H e Q H= ⋅ ⋅ ⋅ (16)

In this case, it is assumed that water is released through the turbines during on-peak hours only. For these types of power plants, downstream re-regulation reservoirs with

Fig. 21. Generating hours for on-peak energy production.

37

sufficient capacity for daily carryover storage are generally needed for providing consistent flow rates downstream. The unchecked Peak Generation Only box instructs MODSIM that releases can be made during off-peak hours. Run-of-the-River Power Plants. Low-head or run-of-the-river projects utilize turbines that require little head for generation. These are modeled in MODSIM by creating a zero capacity reservoir; i.e., a reservoir with no storage capacity, as seen in Fig. 22. Fig. 23 shows the corresponding area-capacity-elevation table, which has only a single entry with a nominal elevation value of 1. Notice in Fig. 24 that power plant elevation is set to 0, which establishes a constant unit valued head and the same power equation as in Eq. 10 is utilized. Efficiency tables and generating hours can be entered as with high-head power plants.

Pumped Storage. Pumped storage projects are indirectly considered by simply increasing the generation hours per period, even beyond the total actual hours in the period. This corresponds to increasing the load factor for the power plant. Although load factor is generally defined as power used/peak power, it can also be defined as on-peak generation hours/total hours. For pumped storage projects, load factor may be > 1.

Fig. 22. Zero capacity reservoir for modeling run-of-the river hydropower plants

Fig. 23. Area-capacity-elevation table for run-of-river hydropower project.

38

In MODSIM, all hydropower plants are assumed to be downstream of storage projects. Hydraulic outlet capacity can be entered as a function of water surface elevation, as shown in Fig. 7, which allows discharges to be restricted to hydraulically feasible flows based on available head on the outlet works. D. Demands Consumptive Demands. MODSIM automatically creates artificial demand links originating at each demand node and accumulating at a single artificial demand node D. The parameters for the artificial demand links are defined as [0, Dit, ck] for artificial

demand link k = [i,D] originating from demand node i, where demands Dit may be

defined as: historical diversions, decreed water rights, predicted agricultural demands based on consumptive use calculations (performed outside the model), or projected municipal and industrial demands. Link weights or costs on the artificial demand links are calculated as:

( )50000 10k ic DEMR= − − ⋅ (17)

where, similar to reservoir priorities, the user selects priorities iDEMR between 0 and

5000, with lower numbers representing higher priorities; i.e., larger negative costs.

Fig. 24. Power plant efficiency for run-of-river projects.

39

Consumptive demand nodes are selected as in Fig. 25, with entry of the demand priority number in the Demand Node Properties form. Clicking the Time Series tab displays the

Time Series form as in Fig. 26 for entry of consumptive demands, and the Groundwater tab, allows entry of infiltration fractions as shown in Fig. 27. Although these demands

Fig. 25. Specification of consumptive demand and priority in Demand Node Properties form.

Fig. 26. Input of demand time series in Demand Node Properties form

40

are designated as consumptive, it is rare that 100% consumption occurs. The fraction of actual water delivery that infiltrates to groundwater is the infiltration fraction, which equals (1 - efficiency). These fractions are allowed to vary seasonally. The remainder of this form provides specification of information related to portions of the infiltration contributing to groundwater and appearing as downstream return flows, possibly at multiple locations. Details on stream-aquifer interactions, including both return flows and stream depletions due to groundwater pumping to supplement water supply to the demand node, are discussed in a subsequent section of this Manual. Instream Flow Demands. MODSIM also provides for non-consumptive flow-through demands that are applied to instream flow uses such as navigation, water pollution control, fish and wildlife maintenance and recreation. The use of flow-through demands for minimum streamflow requirements has two primary advantages: (i) the flow-through demand can be assigned a priority similar to any other demand in the basin, and (ii) simply setting a fixed lower bound on the link corresponding to a minimum streamflow requirement can result in the MODSIM network algorithm prematurely terminating at an

Fig. 27. Entry of seasonal infiltration fractions into the Groundwater form.

41

infeasible solution if there is insufficient flow available to satisfy the minimum streamflow requirement. The flow-through demand can receive a shortage similar to any other demand, depending on the relative ranking of the water right priority assigned to it. An additional advantage of the flow-through demand is that it may be used to divide flow according a predetermined fractional distribution, rather than according to water allocation priorities. As illustrated in Fig. 28, flow-through demands operate by iteratively removing flow as a demand from the network, but then replacing the flow at one or more specified (usually the next downstream) node(s) in the next iteration, which essentially corresponds to a demand with 100% return flow which is unlagged. The superscript k in Fig. 28 represents an iteration counter, since flow-through demand returns must be calculated through a successive approximations procedure. In the first iteration, the instream flow demand D2 is treated as a consumptive demand and flow is delivered according to

priority through solution of the network algorithm. At the next iteration, the flow ( 1)2kq −D

actually delivered in link [2,D] in the previous iteration is then added as an inflow to node 3, and the network is solved once again, but with the bounds on link [2,D] adjusted

to only remove additional flows above what was already flowing (i.e., ( 1)23kq − ) in the

instream flow reach [2,3] in the previous iteration. In this case, link [2,3] is referred to as a bypass credit link, since it is only necessary to augment the streamflow above the

Fig. 28. Illustration of Flow-Through Demands

D

−( 1)2kDq

− − ( 1)

2 23 20, ,kD q c D

1 2 3 ( )23kq

42

current flow level so as to satisfy the minimum streamflow requirement. This solution process continues until successive estimates of returns to node 3 agree. Note that the flow in link [2,3] does not actually represent the total instream flow. Flows leaving node 3 would better represent the actual flows in link [2,3], assuming there are no other demands or inflows at node 3. The output report for demand node 2 properly considers the actual flow in link [2,3] as related to the instream flow requirement. Fig. 29 shows an example of a flow-through demand for a network constructed for use in the Snake River Flow Augmentation Study conducted by the U.S. Bureau of Reclamation and the U.S. Army Corps of Engineers (USBR, 2000). This study examined potential scenarios for reallocation of flows to satisfy instream flow requirements for endangered species in the Snake River basin. A flow-through demand is established at node AndPowr, which accrues flow back to the river reach at node ANDI in the same time step, and without consumptive loss. The specified bypass credit link is _39_ANDI such that the flow-through demand is only used to augment flow in this link in order to satisfy the minimum streamflow requirement. The time series of instream flow requirements is entered at the Time Series tab, and the priority associated with these requirements is entered at the General tab.

Fig. 29. Illustration of flow-through demand in the Upper Snake River basin network.

43

An additional advantage of the flow-through demand is that it may be used to divide flow according to a predetermined fractional distribution, rather than based on demand priorities. This is useful for mutual irrigation companies, or other mechanisms for apportioning flow in a river basin. Flow-through demands are also valuable for model calibration purposes, where a flow-through demand is located at a streamflow gaging station site, and the demands assigned to the flow-through demand node correspond to historical measured flows, which are assigned as the highest priority in the basin. MODSIM parameters and unknown system gains and losses can then be adjusted until measured flows at the gaging station are reasonably matched. Shortage Rules. During higher than normal flow conditions in a river basin, all demands are generally satisfied, whereas during low flow and drought conditions, severe shortages may occur. The priority structure embodied in MODSIM distributes available water supply to high priority uses first. In some river basin systems, the administrative goals are to produce a more equitable sharing of available water during drought. Simply assigning the same priorities to all demands in the basin will not necessarily result in an equitable distribution in a MODSIM solution. Rather, without any priority guidance, MODSIM will produce inconsistent solutions and random distribution of available water. However, similar to use of hydrologic state tables for defining conditional reservoir operating rules, demands can also be conditioned on hydrologic state information, allowing development of shortage rules that attempt to equitably share flow deficiencies among water users during periods of extended drought or low-flow conditions. As seen in Fig. 30, basin-wide demands can be reduced by certain percentages as conditioned on the hydrologic state, allowing more equitable sharing of available water resources during dry periods. Shortage rules are defined in MODSIM by selecting a Hydrologic Table name in the General form within the Demand Node Properties form, as in Fig. 31. Click Apply and OK after selection of the Hydrologic Table, reopen the Demand Node Properties form, uncheck the Varies by Year box, and specify the shortage rule for each period, as in Fig. 32.

Dry

B1τm Bi-1,τm Biτm Bn-1,τm

Ave.

Hydrologic State Rtm

Wet

Dem

and

Di t

Fig. 30. Shortage rules conditioned on Hydrologic state.

44

In some cases, it is desirable to use reservoir operating rules that specify conditional release rules rather than conditional storage target rules for each time period. This is

Fig. 31. Specification of Hydrologic Table for shortage rule.

Fig. 32. Specification of shortage rule in Demand Node Properties form.

45

easily accomplished by specifying an additional flow-through demand node downstream of a reservoir with the desired release levels designated as flow-through demands. These releases can be dependent on storage levels by using the hydrologic state option for flow-through demands

VI. FLOW CONVEYANCE AND ROUTING A. Channel Capacities As seen in the Link Properties form shown in Fig. 33, MODSIM allows users to input constant flow capacity limits for each link, or varying daily, weekly or monthly maximum flow limits for specified variable capacity links. The latter are useful for considering seasonal influences in canal capacities and maintenance schedules. For example, the SouthFork link represented by the form in Fig. 33 is a channel located high in the Rocky Mountains of Colorado. During the winter months, snow and ice restrict flows in the channel. In other regions, seasonal restrictions may be caused by scheduled maintenance, growth of algae, etc. In addition, to variable capacity links, seasonal capacity links can be specified, where the link is constrained to an annual maximum accumulated flow, perhaps based on water rights. Once the accumulated seasonal flows exceed the maximum seasonal capacity, link is effectively turned off, and no further flows through that link are allowed throughout the remainder of the current season. Seasonal flow capacities are then reinitialized to the specified seasonal capacity at the beginning of the next season. All links in the network must be bounded from above and below in MODSIM. Constant minimum flow capacities may be assigned to any link in the network, but care must be taken to avoid infeasible solutions. Improperly assigned minimum and maximum flow capacities on links are the major reasons for network solutions terminating in infeasibility errors. For minimum streamflow requirements, it is best to utilize the flow-through demand construct, as discussed previously. B. Channel Losses: A successive approximations iterative procedure is employed in MODSIM for calculating channel losses, as illustrated in Fig. 34. First, network flows are initiallysolved via the Lagrangian relaxation algorithm with no losses assumed. The losses in each link are

computed by multiplying the loss coefficient by the current flow (0)kq in link k, where the

superscript 0 is an iteration counter representing initial flow in the iterative process calculated without removal of channel losses. This loss is removed during the next iteration by an artificial link terminating at the artificial groundwater GW node with both lower and upper bounds set equal to the amount of loss. The network flow algorithm is then solved again. If current flows in the reach agree with those found in the previous iteration, then convergence has occurred. Otherwise, the procedure is repeated with channel losses defined on the bounds of the artificial link updated to reflect current flows in the real link. This process continues until successive link loss estimates agree within a specified error tolerance.

46

Since the user-specified loss coefficient kcl represents the fraction of flow at the head of

link k that is lost during transition through the link, then during any iteration ℓ, the current

flow ( )kq is the net of losses removed in the channel. That is:

( )( ) . .⋅ − =k ch loss ch losskcl q

or

( )( ) ( ).

1α

= ⋅ = ⋅ −

kk

kch loss k k

clq q

cl (18)

Coefficient αk is then applied to current link flows ( )kq for calculating link losses.

Fig. 34. Specification of channel capacity in Link Properties form.

47

C. Multilinks The multi-link structure of MODSIM allows certain kinds of nonlinear functional relationships to be included in the network model. For example, Fig. 35 shows an example where channel losses in a reach change in a nonlinear relationship with flow rate in the channel. This nonlinear relationship is approximated with a piece-wise linear function, with each linear segment represented as a link connecting the same two nodes i and j. The capacities of each link represent the incremental flow change in that segment, and small unit costs are assigned to each link, with costs increasing with increasing flow. These costs, or perhaps better represented as penalties, are small enough that it is unlikely they will influence the overall distribution of flows in the network. However, they guarantee that link 1, associated with piece-wise segment 1, will fill first, followed by link 2, and finally link 3, as a result of solution by the minimum cost network flow algorithm employed in MODSIM. The accumulated flow in all three links represents the total flow in the channel connecting nodes i and j. It should be noted that any costs assigned to network links are treated directly as costs, and not priorities which are translated into costs using Eq. 4 or Eq. 17. Clinking Convert to Multilink on the context menu for any link allows creation of any number of links connecting the same two nodes, as in Fig. 35. Editing of the parameters of each link is easily accomplished, such as shown in Fig. 36 for link 1 in the above

Fig. 34. Successive approximations procedure for channel loss calculations.

Channel Loss Link j

( )kjq

α α− −

⋅ ⋅( 1) ( 1), ,0k kj jj jq q

link loss

( )α =−1

jj

j

cl

cl

iteration no.

GW

48

example, where the flow increment for that segment of the piece-wise linear function is set as the maximum flow in that link, the link cost is set to 0, and the channel loss coefficient (slope of linear segment 1 in Fig. 35) is specified under the Channel Loss tab. For certain problems where it is desirable to include pumping costs, MODSIM allows direct entry of unit costs in the Link Properties form for any link in the network. Negative costs can be entered to represent benefits, such as from low head hydropower production, since MODSIM solves a minimum cost network flow optimization problem. If these costs are nonlinear functions of flow in the link, the multilink method described previously for nonlinear channel loss functions can be similarly applied. In this case, the cost functions must be convex (i.e., with increasing slope) and benefit functions (represented as negative costs) must be concave (i.e., decreasing slopes in order to insure correct minimum cost solutions by the network flow optimization algorithm. Costs or penalties (positive or negative) can be directly assigned to any link by the user to discourage or encourage, respectively, flow in that particular link according to predefined operational criteria. It must be remembered, however, that if water right priorities are included in the network, then any link costs introduced by the user must be set at small

1000 1800 kq

.Ch Loss

1kcl

2kcl

3kcl

i j

[ ]0,1000,0

( )0, 1800 1000 ,1 −

[ ]0,9999,2

[ ], ,k k kl u c

1kcl

2kcl

3kcl

Fig. 35. Illustration of piece-wise linearization of channel loss as nonlinear function of flow using a multilink.

49

relative values that will not disrupt the distribution of flows according to the water right priorities. D. Channel Routing For simulation of daily streamflow, it may be necessary to consider channel routing. This is accomplished in MODSIM by designating a network link as a routing link. Inflow to this link is distributed over time in accordance with routing coefficients calculated by MODSIM using the Muskingum formula. Alternatively, the user may directly input any desired routing coefficients and lagging factors. An iterative process similar to calculation of flow-through demands is employed, except that returns to the channel are distributed according to the routing coefficients over the current and future time steps. Assuming that the flow entering routing link k during time step t is ktq , then routed

outflow from link k is:

Fig. 36. Specification of flow increment, unit cost and channel loss coefficient for link 1.

kcl

50

0 1 , 1 2 , 2− −′ = + + +kt kt k t k tq c q c q c q (19)

where ′ktq is the ordinate of the outflow hydrograph at time t; , τ−j tq is the ordinate of

the inflow hydrograph at time t – τ for τ = 0,1,…; and 0 1, ,c c …are routing coefficients.

The routing coefficients may be calculated by MODSIM based on the Muskingum routing equations by selecting Model Generated under Groundwater Lag Factors in the Network Settings form, as shown in Fig. 37. Selecting the User Generated option and specifying the Maximum Number of Lags assumes that users will enter the routing coefficients directly.

The Muskingum routing coefficients are calculated by MODSIM as follows:

( )( )( )0

2

2 1

∆ −=

− + ∆t XK

cK X t

(20)

( )( )( )

( )( )2 1 2

2 1

− + ∆ − ∆′ =

− + ∆

K X t tc

K X t (21)

( )( )( )1 0

2

2 1

∆ +′= ⋅ +

− + ∆t XK

c c cK X t

(23)

1 for 2−= ⋅ ≥i ic c cc i (24)

where users enter weighting factor X as a dimensionless number usually between 0 and 0.5 representing the relative influence of the inflow in determining the prism storage volume in reach k. Parameter K is in units of days (or fraction of a day) in this case, representing travel time through the reach, and t∆ is generally set to 1 for daily time increments. To avoid negative coefficients, the Muskingum parameter K should be within the following limits:

Fig. 37. Selection of User Generated groundwater lag factors or Model Generated (Muskingum method).

51

( )( ) 22 1

t tK

XX

∆ ∆< ≤⋅ −

(25)

An iterative process is employed in MODSIM for routing daily flows, as illustrated in Fig. 38, where the superscript ℓ is an iteration counter. At iteration ℓ = 0, an initial MODSIM

solution is found that temporarily ignores flow routing. This results in flow (0)ktq in

routing link k during current time t. An artificial link is then created that removes the portion of this flow appearing downstream in future time steps, with the remaining flow passing downstream representing the portion of the routed link outflow that occurs in the current time step t. The routed outflow appearing in future time steps is then placed back into a return node immediately downstream of the ending node of the routing link (node n+1 in Fig. 38). Again, these flows will not appear until future time steps. Notice that the link parameters on the artificial link specify that exactly the flow specified by the upper and lower bounds must be removed from the river reach and subsequently added

back in downstream. The iterative process continues until iteration ℓ when ( ) ( 1)−≅kt ktq q

within a user specified convergence tolerance.

Fig. 39 gives an example of data entry for a routing link in the daily operations network for the Geum River basin, Korea. It is important to note that the downstream node for a routing reach should not be assigned as a demand node since it will interfere with the routing calculation. In addition, routing links should not be connected in direct succession in the network configuration. Fig. 40 shows entry of user specified lag factors or routing coefficients in the Link Properties form.

Fig. 38. Successive approximations procedure for channel routing.

( ) ( )− − − ⋅ − ⋅ 0 0( 1) ( 1)1 , 1 ,0kt ktc q c q

( )ktq

Routed Flows from Previous Periods

Iteration No.

τ ττ

−=∑ ,1,2,...

k tc q

GW

Routing Link

52

Fig. 40. User specification of routing coefficients.

Fig. 39. Selection of Channel Routing link in Link Properties form.

53

E. Backrouting in Daily Reservoir Operations Overview. The channel routing procedure in MODSIM temporarily disconnects the network at each routing link due to the iterative solution procedure that is employed. Because of this, a lower priority water use upstream of a routing link may receive flows in lieu of deliveries to higher priority downstream uses. To overcome this problem, an innovative backrouting procedure is incorporated in MODSIM to insure that water deliveries occur in the correct priority. A look-ahead approach is adopted whereby water delivery decisions for time step t are based on knowledge about future water system requirements using several network runs over future concurrent time steps. Downstream demand time series are backrouted upstream to represent flows in the current time step t required to pass through the routing link to satisfy future downstream water requirements without unnecessary shortages and spills. General Procedure. MODSIM implements network optimization using the Lagrangian relaxation solver, with each time step solved as a separate problem, since network optimization is utilized in MODSIM to simulate the allocation of available water resources according to prespecified priorities and rules. The network optimization progresses from the starting simulation date to the ending simulation date without any knowledge about future events or demands in the system. As described previously, channel routing in MODSIM over daily time steps is accomplished by defining a routing link with appropriate Muskingum routing coefficients or user-defined lag coefficients. The flow routing is accomplished by removing a portion of the flow from the upstream node defining the routing link, which is then returned to the downstream node of the routing link in future time steps according to the routing coefficients. Iterations in the current time step are repeated until successive network solutions match according to prespecified convergence criteria. The normal streamflow routing procedure employed in MODSIM will produce correct solutions as long as there is sufficient water to satisfy all demands, whether they are of low or high priority. Difficulties arise when there is insufficient water available to meet all demands, and priorities exist on allocation of water. Under water shortage conditions and priroties on demands, routing time steps longer than one day can cause downstream demands to pull water from upstream reservoirs, although they do not receive this water immediately. This causes unnecessary releases of additional water from upstream reservoirs that are in excess of downstream demands. A backrouting methodology has been implemented in MODSIM to overcome the problem of excessive reservoir drawdown associated with longer routing periods. In this approach, water delivery decisions for time step t are based on knowledge about future water system requirements that are calculated using several network runs over future concurrent times. The concept of concurrent time networks is based on the fact that in time step t, only a fraction of the water available at any location in the network will actually reach the farthest downstream region of the network during the same time step. For example, in Fig. 41, none of the water available in link A will reach the farthest downstream node during period t, and only 50% of the water flowing in link B will reach the farthest downstream node during the current time step, with the remaining 50% of the

54

Fig. 41. Simple real-time K- MODSIM routing network.

Routing Link (3)

water arriving at the most downstream point in time t+1. A concurrent time network that represents only the water available at the same time in the farthest downstream region is called a downstream time network. The downstream time network is constructed by transforming demands, inflows, groundwater return flows and reservoir storage targets to the corresponding portions of the flow available. Downstream time networks for current and future time t+τ (τ = 0,…,n), where n is the maximum number of time lags, combine portions of previous time flows that reach the most downstream region in period t+τ, and the portion of the water available in the system at time t+τ able to reach the farthest downstream node. Mathematical calculation of the downstream time network flows is accomplished by defining regional routing coefficients resulting combination of the multiple routing effects. Routing links in the MODSIM network essentially divide the system into routing regions (Fig. 41) such that flows in each region are considered to be concurrent in time. The combination of routing link coefficients from the farthest downstream region that are backrouted to each upstream region allow calculation of regional routing coefficients jr that transform

the flow time series in region j to an equivalent time series in the farthest downstream region time frame. In this example, Region 1, the farthest downstream region, has a regional routing coefficient vector composed of the value 1 in the first element, followed by 0 values in the remaining elements, since all the water in Region 1 will flow to the farthest downstream region (i.e., itself) in the current time step.

10 10

11 121 1

1 1

1 10 0

0 0n n

c rc r

c r

= = = =

c r

and routing coefficients for next upstream region (Region 2) are represented as:

20

212

2n

cc

c

=

c

55

Fig. 42. Routing regions in K-MODSIM Network.

Routing Link (3) c3 = [0,1]

Routing Link (2) c2 = [0.5,0.5]

REGION 3 N

REGION 2

REGION 1

where n represents the position of the first zero valued element in the vector 2c after the

lexicographically last positive-valued element (i.e.,

2, 1 0nc − > ; 2 0nc = ). The regional routing

coefficients for Region 2 are then calculated as a convolution process:

10 20 20

11 10 21 212 1 2

1 1, 1 10 2 2

0 0 00 0

0

n n n n

r c rr r c r

r r r c r−

= ⋅ = ⋅ =

r R c

For the example shown in Fig. 42, 2r would be

calculated as:

2 1 2

1 0 0 0.5 0.50 1 0 0.5 0.50 0 1 0 0

= ⋅ = ⋅ =

r R c

This convolution process continues upstream for Region j:

1,0 0 0

1,1 1,0 1 11

1, 1, 1 1,0

0 0 0

0 0

0

(for 2,..., )

j j j

j j j jj j j

j n j n j jn jn

r c r

r r c r

r r r c r

j N

−

− −−

− − − −

= ⋅ = ⋅ =

=

r R c

where N is the total number of the regions and n represents the position of the first zero valued element in the vector jc after the lexicographically last positive-valued element

(i.e., , 1 0j nc − > ; 0jnc = ), or the position of the first zero valued element in the regional

routing vector 1j−r (i.e., 1, 1 0j nr − − > ; 1, 0j nr − = ), whichever is larger. For the simple

example of Fig. 42:

3 2 3

0.5 0 0 0 0 00.5 0.5 0 0 1 0.50 0.5 0.5 0 0 0.50 0 0.5 0.5 0 0

= ⋅ = ⋅ =

r R c

56

Let jtq represent the time series of inflows, demands or groundwater return flows in

Region j. These time series are routed to the farthest downstream region time using the computed regional routing matrices jR as follows:

0

1 1,0 , 1 , 1

1, 1, 1 1,0 , ,

0 0 0

0 0

0

j jt jt

j j j t j tj j j

j m j m j j t m j t m

r q q

r r q q

r r r q q

− + +

− − − − + +

′ ′ ′ = ⋅ = ⋅ = ′

q R q

where m represents the position number of the first nonzero element in the regional routing vector associated with the farthest upstream region N. That is, m is defined such

that for ( )0 , 1 , 1,..., , , ,...,N N m Nm N m Nmr r r r r− + , elements 0 , 1 0, 0N N m Nmr r r−= = = > . The

farthest upstream region is used for defining m since the multiplied effects of downstream routing will be maximized in the farthest upstream region, which determines the number of future time networks that will need to be solved. The downstream time series represent concurrent flows in the downstream time t + τ, where τ is the index of future time steps. Fig. 43 shows successive downstream network for two time steps. Notice that there are no routing links in the downstream networks and all water users and reservoirs compete for concurrent water in a continuous network. At every sequential real time step t, a series of m+1 downstream time networks over the current and future time steps are run to calculate current and future water requirements

Fig. 43. Downstream Time Networks Solved at Each Real-Time Step t

Downstream Time t +1

′ = ⋅ 31

Node InflowAt Atq q r +

+

′ = ⋅⋅

, 1 32

, 1 31

Node Inflow

+ A t At

A t

q q rq r

′ = ⋅ 21

Node InflowBt Btq q r +

+

′ = ⋅+ ⋅

, 1 22

, 1 21

Node Inflow

B t Bt

B t

q q rq r

′ = ⋅Node Demand

1Dt Dtq q + +′ = ⋅ + ⋅, 1 , 1

Node Demand0 1D t Dt D tq q q

Downstream Time t

57

for the routing links in the real network. The flows ( ), 0,1,..., )τ τ+′ =j tq m calculated in

these network solutions represent flows in routing link j, but translated to downstream region time t+τ. These flows are now backrouted to time t using the inverse of the regional routing coefficient matrix jR as follows:

( )

1

1 , 1 , 1 , 1

, , 1 , ,

ˆ0 0 0

ˆ0 0ˆ

0ˆ

j j j

j j j jj j j

j m j m j j m j m

r q q

r r q q

r r r q q

−

− + + +

+ + − + +

′ ′′ ′ ′′ ′′ ′= ⋅ = ⋅ = ′ ′′

q R q

where represents the lexicographically first nonzero element in the regional routing vector jr . Defining the index truncates the regional routing matrix by removing 0

valued columns and rows in the matrix. This prevents the routing matrix from being singular and therefore allows calculation of the matrix inverse. Since there is no routing occurring prior to time step t+ (i.e., the regional routing coefficients = 0 for those time steps), then , for 0,..., 1τ τ τ′′ ′= = −j jq q . Therefore, the final backrouted flows

translated to current time t for routing link j are:

0 0

, 1 , 1

, ,

ˆ

ˆ

j j

j jj

j j

j m j m

q q

q q

q q

q q

− −

+ +

′ ′′

′ ′′ ′′ = =

′′ ′′ ′′ ′′

q

In effect, the backrouted flows 0jq′′

represent flows in the current real-time step t required in the routing link downstream of Region j to meet future water requirements without unnecessary shortages and spills. Note that even though the entire backrouted vector j′′q is calculated, only the first

element is actually used in time step t since future backrouted flows will be recalculated when the simulation moves to the next time step t+1. Invoking backrouting only requires selection of Backrouting under Extensions in the MODSIM main menu item (Fig. 44). The same information as required for channel routing is necessary for the backrouting procedure.

Fig. 44. Selection of backrouting as a K-MODSIM extension.

58

Backrouting Example. This example attempts to illustrate the advantages of using the backrouting procedure to insure that flow routing does not interfere with the priorities associated with water allocation in the network, particularly during drought or low-flow conditions. Fig. 45 shows a simple network with a low priority reservoir node as a limited source of water, and two demands separated by a routing link. The node “dem2” has a higher priority over the node “dem1” in the water delivery, but both have the same water demand of 100 units. This example represents a case with time lags of more than one day, where the routing coefficients are zero in the first time step, 0.20 in the second, and 0.80 in the third. In order to precondition the network for previous flow conditions due to flow routing (i.e., prior to the starting time step), 80 units of flow are returned to node ms2 in the first time step, since the real system will rarely be completely dry in the downstream section. The lack of returning routed flows in the first time step causes excessive reservoir releases, thereby moving the solution farther from the real operation. This example demonstrates the look-ahead capabilities of the back routing algorithm in the way that MODSIM solves the network. The physical channel routing solution (i.e., without backrouting) releases all water from the reservoir in an attempt to satisfy the senior demands downstream of the routing link. The flow in the link going out the reservoir is shown in Fig. 46.

Fig. 45. Backrouting example.

Routing Link

[0.2,0.8]

Routing Link

[0.2,0.8]

Fig. 46. K-MODSIM Results for Streamflow Routing Case Showing Large Reservoir Releases

59

It can be seen in Fig. 47 that the senior demand receives water mainly from flows added to precondition the network for routed flows prior to the current simulation period. Large shortages are observed for the case where the physical channel routing option is used.

In this solution, the junior demand receives no water, and most of the water is spilled at the downstream sink node, as seen in Fig. 48.

Fig. 47. Shortages to senior demand under the physical channel routing solution.

Fig. 48. Large downstream spills in the physical channel routing solution.

Routed Flow

Routed Flow,

60

Fig. 49 shows that under the backrouting solution, the model produces the exact reservoir releases needed to meet demands over the future time steps. In contrast with the solution

under physical channel routing, the junior demand (dem1) is able to receive sufficient flows to satisfy the demands through the first four time periods, as seen in Fig. 50.

Fig. 50. Correct reservoir releases under the backrouting solution.

Routed Flow

Routed Flow,

Fig. 49. Demand satisfaction for low priority demand with backrouting.

61

Demands for the high priority demand node (dem2) are generally satisfied under backrouting, although some shortage occurs in the first period since it is physically impossible for the senior demand to receive water from the reservoir until the second time step due to the time lags (Fig. 51). The flow through the routing link under the backrouting procedure is shown in Fig. 52. The benefits of backrouting are demonstrated by the fact that no spills occur with this solution.

Fig. 51. Demand satisfaction for the senior demand (dem2) under the backrouting solution.

Fig. 52. Flow through the routing link for the backrouting solution.

Routed Flow

Routed Flow,

62

VII. STREAM-AQUIFER MODELING COMPONENTS A. Analytical Equations The stream-aquifer module within MODSIM calculates reservoir seepage, irrigation infiltration, pumping, channel losses, return flows, river depletion due to pumping, and aquifer storage. Stream-aquifer return/depletion flows are simulated using response coefficients calculated using the one dimensional equations developed by Glover (1977). These are similar to groundwater response coefficients estimated from the stream depletion factor (sdf) method of Jenkins (1968). Alternatively, as described in Fredericks, et al. (1998), response coefficients can be generated from the three-dimensional finite difference groundwater model MODRSP/MODFLOW (Maddock and Lacher, 1991) and read into MODSIM from external data files. This allows response coefficients to be calculated based on spatially distributed aquifer characteristics and complex boundary conditions. Details on calculation of return flows, stream depletion from pumping, and canal seepage in MODSIM can be found in Fredericks, et al. (1998).

VIII. Storage Accounts, Exchanges and Water Banking MODSIM includes important administrative features including storage contract arrangements such as accrual rights, ownership contracts or agreements, water service contracts, and rental pool or water banking. Fig. 53 highlights Arrowrock Reservoir in the Boise River basin, Idaho, as a storage account reservoir. Although depicted as an offstream reservoir, in reality it is an onstream reservoir that includes several storage accounts with group ownerships. The reservoir bypass link GainArkk__2 represents any flows passing directly through the reservoir. The reservoir outflow link Arrowrok__2 conveys releases from the storage accounts to downstream storage ownerships or for exchange purposes. Accrual to the storage account reservoir occurs through natural flow links originating from the nonstorage node GainArkk which are governed by natural flow rights and decree amounts. Each storage account in the reservoir is represented by a unique accrual link, with this account representing space in the reservoir with a priority date stated in the storage right permit granted by the state water resources department. Once flow has accrued to the storage accounts, releases are governed by the needs of the owners of the storage contracts. Normally, the sum of all storage accounts should not exceed the total active capacity of the reservoir, although there are exceptions. Unused carryover accrual from the previous year may or may not be allowed, at the discretion of the model user. For storage account reservoirs, evaporation loss is distributed as a negative accrual to the storage accounts in proportion to the current total accrued storage in the account. Several parallel links conveying flow to a demand node may be defined when the demand owns several direct diversion or natural flow rights in the river. In these cases, the link capacity corresponds to the decreed water right amount. Additional links can be created and designated as storage ownership links, where flows are made available through releases from storage accounts in reservoirs accruing water from decreed storage rights.

63

MODSIM performs a two step process for river basin management studies that include storage accounts and ownership: 1. The natural flow step distributes flows based on water right priority, including accruals to the storage accounts. For the latter, the accrual links are constrained to a seasonal capacity corresponding to the size of the storage account. During this step, the reservoir outflow links are turned-off to prevent any releases from storage accounts. Storage ownership links conveying storage account releases to downstream storage account owners, such as shown in Fig. 52, are also temporarily closed during the natural flow step. This step insures that all legal entitlements are correctly maintained. 2. In the storage step, all flows in the natural flow and accrual links are constrained to exactly deliver the flows allocated during the natural flow step. The reservoir outflow links and storage ownership links are now opened and the network is solved again to provide the physical simulation of storage accounting in the basin. Releases from storage accounts satisfy any remaining demands of the storage ownerships after exhausting their direct flow or natural flow rights. However, MODSIM maintains a strict accounting to make sure that drafts are made only if storage is available in that account. In this process, storage owners may have accounts in several reservoirs, with priorities as to which to receive first governed by assignment of system numbers. In addition, individual storage

Fig. 53. Example of Arrowrock Reservoir, Boise River basin, as a storage account reservoir.

64

accounts may be owned by ownership groups, with releases equitably distributed to the individual owners in that group. MODSIM supports development of exchange agreements between users to encourage efficient use of water resources. For example, storage account holders may be unable to physically receive releases from the reservoir in which the account is held. During the storage step, when the owner’s storage ownership link is opened, natural flow will be diverted out of priority in lieu of reservoir releases from the owner’s storage account delivered downstream to senior water right holders. Exchanges also improve the efficiency of reservoir operations by separating theoretical accounting from physical storage in the system. It may be advantageous to physically store water in an upstream reservoir and account for the storage fill as if it occurred in the downstream reservoir. That is, water belonging to a contract holder in one reservoir may be physically held in another. MODSIM insures that at a user-specified date, the theoretical and physical storage accounts are balanced in the river basin. Each link in a MODSIM network has a field for optionally specifying an exchange limit link. This field is an example of the watch logic incorporated in MODSIM whereby the flow passing through a specified exchange limit link in a previous iteration is watched, with the upper bound on the link with the specified exchange limit link is set to that flow value. Watch logic is a tool to simulate explicitly defined exchanges where the allowed flow at some location in the network is contingent on flow elsewhere in the network. Similarly, fields are available in the demand node properties data form for specifying exchange credit links or exchange credit nodes. MODSIM applies watch logic such that the flow passing through a specified exchange credit link or flowing into a specified exchange credit node in a previous iteration overrides the assigned demand for that node. Rent pool can simulate water banks or water service contracts by allowing storage ownership to be temporarily transferred to another demand. If a demand has more water entitlement than needed in a high runoff year, the demand can contribute part or all of the ownership accrual to the rent pool. Other demands requiring more water than entitlements dictate can subscribe to the rent pool in a given year (Labadie, et al., 2002).

IX. MODSIM 8.0 OUTPUT PROCESSING AND CUSTOMIZATION A. Graphical Output and Frequency Analysis After running the MODSIM network solver, results can be saved for all nodes and links, or for a group of nodes and links selected prior to model execution. MODSIM output files are comma-and-quote delimited ASCII text files that are readily imported into spreadsheet software. MODSIM displays graphical output results in the GUI using the freeware graphing package TeeChart (Steema Software, SL). In addition to TeeChart, the PISCES graphing package developed by the U.S. Bureau of Reclamation, Pacific Northwest Region, can display MODSIM output files. As seen in Fig. 54, PISCES provides powerful tools for comparative evaluation of several river basin management

65

scenarios, as well as performing frequency analysis and plotting flow-duration curves. B. Water Quality Management The customization features of MODSIM and the ease of linking it with other models have allowed access to modeling capabilities not directly available in MODSIM. For example, de Azevedo, et al. (2000) integrated MODSIM with the QUAL2E-UNCAS stream water quality model for evaluating strategic planning alternatives for meeting transbasin diversion requirements for the city of Saõ Paulo, intrabasin water supply needs, and acceptable water quality according to various reliability criteria. Batch processing was applied to automating the conversion and transfer of MODSIM network flow results for input to QUAL2E-UNCAS. This environment allowed adjustment of operating targets and priorities in MODSIM to achieve integrated water quantity and quality objectives. The U.S. Geological Survey developed a DSS called SIAM (System Impact Assessment Model) for the Klamath River basin, Oregon and California, that links MODSIM with the HEC-5Q reservoir water quality model, an aquatic habitat model, and the SALMOD fish production model. SIAM was developed to explore the potential for changing system operations in the Klamath River Basin to improve summer/fall water quality conditions for the benefit of declining anadromous fish populations (Campbell, et al., 2001). The SIAM GUI allows users to modify operational data for an existing MODSIM network of the Klamath River basin, or allows experienced users direct access to the MODSIM GUI for modifying network features.

Fig. 54 Example display of flow frequency distribution (or flow duration) curves with the PISCES graphing package for the Upper Snake River Basin

66

The Colorado Water Resources Research Institute sponsored application of MODSIM to the Lower Arkansas River Basin below Pueblo Dam for identifying opportunities to improve water quality in the Lower Arkansas River Basin through conjunctive use of groundwater and surface water by linking MODSIM with the QUAL2E stream quality model and a groundwater return flow salinity model (Dai and Labadie, 2001). This application customized MODSIM to execute QUAL2E as MODSIM is running. C. Economics-Driven Simulation Although MODSIM is often utilized for water rights evaluation in river basin management, the U.S. Bureau of Reclamation, Sacramento, California sponsored application of MODSIM to the San Joaquin River Basin to investigate the use of economic-based strategies such as increased water prices, tiered water pricing, changes in San Joaquin River environmental flows, and changes in reservoir operations to improve water management. This study made effective use of the scripting capabilities of MODSIM for modeling the complex water pricing structures in the basin (Leu, 2001).

67

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Maddock III, T. and Lacher, L., “MODRSP: A Program to Calculate Drawdown, Velocity, Storage and Capture Response Functions for Multi-aquifer Systems,” HWR Report No. 91-020, Department of Hydrology and Water Resources, University of Arizona, Tucson, Ariz., 1991.

69

Miller, S., Johnson, G., Cosgrove, D., and Larson, R., “Regional Scale Modeling of Surface and Groundwater Interaction in the Snake River Basin, Journal of the American Water Resources Association, Vol. 39, No. 3, pp.517-528, 2003.

Salas, J., Saada, N., Chung, C-H, Lane, W. and Frevert, D., “Stochastic Analysis, Modeling and Simulation (SAMS), Users Manual,” Deptartment of Civil Engineering, Colorado State University, Ft. Collins, Colo., 2002.

Shafer, J. and Labadie, J., “Synthesis and Calibration of a River Basin Water Management Model,” Completion Report No. 89, Colo. Water Resources Research Institute, Colorado State University, Ft. Collins, Colo., 1978.

Sprague, R. and Carlson, E., Building Effective Decision Support Systems, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1982.

State of Colorado, “Documentation for STATEMOD Water Rights Model,” Denver, Colo., 1999.

State of Wyoming, “Wind River WIRSOS Users Manual,” State Engineers Office, Cheyenne, Wyo., 1992.

Stillwater, L., “Little Butte and Bear Creek Surface Water Distribution Model,” Draft – Model Version, U.S. Bureau of Reclamation, Pacific Northwest Region, Boise, Idaho, 2003.

Stillwater, L., “Cascade Drawdown Studies – Draft Model Documentation and Study Results,” U.S. Bureau of Reclamation, Pacific Northwest Region, Boise, Idaho, 2004a.

Stillwater, L., “Water Distribution Model – Tualatin Project BA,” U.S. Bureau of Reclamation, Pacific Northwest Region, Boise, Idaho, 2004b.

Salem, T. and Labadie, J., “Optimal Conjunctive Use of Surface and Groundwater Resources in the Lower Nile,” Planning Studies and Models Project, Ministry of Public Works and Water Resources, Cairo, Egypt, 1995.

U.S. Bureau of Reclamation, “River and Reservoir Simulation of the Snake River: Application of MODSIM to the Snake River Basin,” U.S. Department of the Interior, Bureau of Reclamation, Pacific Northwest Region, Boise, Idaho, 2000.

Weiss, P. J. Labadie, and M. Baldo, “Environmental Impact Evaluation Using a River Basin Network Flow Model,” Proceedings of the 24th Annual Water Resources Conference, D. Merritt (ed.), Water Resources Planning and Management Division, ASCE, Houston, Tex., pp. 74-81, Apr. 6-9, 1997.

Wurbs, R. A. and W. B. Walls, “Water Rights Modeling and Analysis,” Journal of Water Resources Planning and Management, Vol. 115, No. 4, pp. 416-430, 1989.

70

Appendix A LAGRANGIAN RELAXATION ALGORITHM FOR SOLVING

MINIMUM COST NETWORKS

A.1 Karush-Kuhn-Tucker (KKT) Conditions1 Consider the following general optimization problem: Primal: subject to:

where functions ( )⋅f and ( )⋅ig are real-valued and differentiable. The Karush-Kuhn-

Tucker (KKT) conditions state that if *x solves the above problem, then the following conditions must hold:

where the Lagrangian function is defined as:

These are only necessary conditions for an optimal solution; that is, they may also be satisfied at points other than *x (e.g., local minima, local maxima, or saddle points). Note that the complementary slackness conditions are automatically satisfied for problems with equality constraints, and are therefore not stated in the above conditions. A.2 Saddle Point Conditions

The following conditions are sufficient conditions; that is, if they are satisfied for a particular point *x , then *x must be the solution to the primal problem. They are not, in general, necessary for an optimal solution *x to the primal problem. In some situations, they may not hold for *x . These conditions are called saddle-point (SP) conditions:

1 Bazarra, M., H. Sherali, and C. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York, 1993.

min ( )fx

x

1

( ) 0, 1,...,

( ,..., )

i

nn

g i m

x x E

= =

= ∈

x

x

* * *1

*

* *x

( ,..., ) such that:

(i) ( ) 0, 1,..., [feasibility]

(ii) ( , ) [stationarity]

m

ig i m

L

λ λ∃ =

= =

∇ =

λx

x λ 0

* * * * *

1

( , ) ( ) ( )m

i ii

L f gλ=

= +∑x λ x x

* * * *1

*

* *

*

and ( ,..., ) such that:

(a) ( ) 0, 1,..., [feasibility]

(b) minimizes ( , ) [optimality]

solves the primal problem

m

i

p p

g i m

L

∃ =

= =

IF x p

x

x x p

THEN x

71

proof: From (b)…

* * *( , ) ( , )L L≤x p x p and from (a)…

* * *

1

( ) ( )m

i ii

f p g=

+ ∑x x0

*

1

( ) ( )m

i ii

f p g=

≤ + ∑x x0

Therefore… *( ) ( ), subject to (a)f f≤x x

or *x solves the Primal. || If the Primal is a convex programming problem, the KKT and SP conditions are exactly equivalent (since (ii)⇔ (b)). In this special case, these conditions are both necessary and sufficient. Definition: the iλ in the KKT conditions are called Lagrange multipliers.

Definition: the ip in the SP conditions are called generalized Lagrange multipliers, dual

variables, or dual prices. Note that Lagrange multipliers and dual variables are not necessarily equivalent, unless the KKT and SP conditions are both necessary and sufficient. A.3 Dual Problem

The SP conditions immediately suggest a hierarchical strategy if (b) is more easily solved than the primal problem (it is always easier to solve an unconstrained problem than a constrained one). Consider the following 2 level structure: A point * *( , )x p that satisfied the SP conditions is called a saddle-point. It can be shown that:

* * * *( , ) ( , ) ( , )L L L≤ ≤x p x p x p

This problem manipulation is called dualization since the saddle point conditions state that: p

x

( , )L x p

Saddle Point

Check if (a) and (b) satisfied.

If not, adjust p further

Solve condition (b):

min ( , )Lx

x p

2nd level:

1st level:

*( )x p p

( )* * *( , ) max ( , )

max min ( , )

=

=

L L

Lp

xp

x p x p

x p

72

We can define the dual function as:

and the dual problem is:

If a saddle point exists, this dual problem will yield *p , along with the *x that minimizes

*( , )L x p , which solves the primal problem. It can be proved that the dual function ( )h p is always a concave function with respect to the dual variables p , even if the original primal problem is nonconvex. This means that a global optimum can always be found when solving the dual problem. Unfortunately, for nonconvex problems, there may be some situations where a duality gap exists. That is, even though *p solves the dual problem, the resulting solution *x does not solve the

original primal problem. From the saddle point conditions, * *( ) ( )h f=p x if there is no

duality gap. If a duality gap exists, however, then * *( ) ( )h f<p x and the difference between these two values represents the extent of the gap. If the original primal problem is convex, then it is guaranteed that no duality gap exists. A.4 Example

Consider the following problem:

subject to: The optimal solution to this problem is . Note that we have added lower and upper bound constraints on the variables, but we will see that the dual problem is still easily solved in this case. The dual function for this problem is: Notice that we have only included the equality constraint in the Lagrangian function, and not the lower and upper bound constraints on the variables. These bounds are explicitly included in the above minimization. On the other hand, the equal constraint, by being included in the Lagrangian function, is no longer an explicit constraint in the dual problem, and has been relaxed. We will attempt to find the optimal dual price p that indirectly results in satisfaction of this constraint.

{ }

{ } { }1 2

2 21 2 1 20 2

( 1,2)

2 21 1 2 20 2 0 2

( ) min ( 1) ( 2) ( 2)

min ( 1) + min ( 2) 2

jxj

x x

h p x x p x x

x p x x p x p

≤ ≤=

≤ ≤ ≤ ≤

= − + − + + −

= − + − + −

* (0.5,1.5) and min 0.5T f= =x

1 2 2

0 2, 1, 2j

x x

x j

+ =≤ ≤ =

2 21 2min ( 1) ( 2)x x− + −

x

*( ) ( , )h L=p x p

max ( )hp

p

73

We could use any convenient minimization technique to solve the above separable optimization problems. In this example, it is convenient to use calculus, and the optimal solutions can be expressed parametrically as functions of the dual variable p:

* *1 2( ) 1 ; ( ) 2

2 2

p px p x p= − = −

Inserting these solutions into the dual function gives:

2

( )2

ph p p= −

which is, as expected, a concave function of p . Again using calculus, solution of the dual problem:

2

max 2p

pp −

gives * *1 and ( ) 0.5p h p= = . Instead of using calculus, any efficient gradient-type search technique could be used for more complex problems. Inserting this solution into the above parametric optimal solutions for x(p) gives * (0.5,1.5) T=x . This solution happens to satisfy the bound constraints on the variables, and also satisfies the equality constraint. This is the correct optimal solution and indeed * *( ) ( ) 0.5h p f= =x at the saddle point. A.5 Lagrangian Relaxation Algorithm for Solving Minimum Cost Networks2

Problem Formulation. The minimum cost network flow algorithm employed in K-MODSIM solves a dual problem similar to that described above, but specialized to take advantage of the linear network structure of the problem. In this formulation, link or arc (i,j) in K-MODSIM is designated by the node pair (i,j) representing the beginning and ending nodes of the link, respectively. This notation implies one unique node pair for each link, and is used for notational convenience only in the following development. The algorithm is actually capable of considering multiple links for the same node pair. The objective function is:

( , )

min ij iji j A

c x∈∑

x

subject to:

{ } { }( , ) ( , )

0 , 1,...,

ij ij ij

ij jij i j A j j i A

l x u

x x i N∈ ∈

≤ ≤

− = =∑ ∑OUT IN

where ijx represents the flow rate in arc (i,j) with link parameters , ,ij ij ijl u c , ijc is the

cost per unit flow for arc (i,j), A is the set of all links or arcs in the network, N is the total

2 Bertsekas, D. and P. Tseng, “RELAX-IV: A Faster Version of the RELAX Code for Solving Minimum Cost Flow Problems,” Completion Report under NSF Grant CCR-9103804, Department of Electrical Engineering and Computer Science, MIT, Cambridge, Mass., 1994

i j ijx j’

'j ix

74

number of nodes, and ijl and iju are the link flow lower and upper bounds, respectively.

A transformation can be performed to remove the lower bounds from this problem. Let or ij ij ij ij ij ijx x l x x l= − = +

( , )ij ij iju u l i j A= − ∀ ∈

The transformed objective function is now formulated as:

( , )

min ij ij iji j A

c x l∈

+ ∑x

Since the constant term can be removed, the objective is:

( , )

min ij iji j A

c x∈∑

x

subject to:

{ } { }( , ) ( , )

0 ( , )

0 1,...,

ij ij ij ij

ij ij ji jij i j A j j i A

x u u l i j A

x l x l i N∈ ∈

≤ ≤ = − ∀ ∈

+ − + = = ∑ ∑

or

{ } { }( , ) ( , )

, 1,...,ij ji ij i j A j j i A

x x s i N∈ ∈

− = =∑ ∑

where

{ } { }( , ) ( , )

, 1,...,i ji ijj j i A j i j A

s l l i N∈ ∈

= − =∑ ∑

In this formulation, all link parameter data [uij , cij] and si are assumed to be integer values. A.6 Lagrangian Relaxation Algorithm

The solution to this problem is based on a Lagrangian relaxation algorithm developed by Bertsekas (1991). Introducing generalized Lagrange multipliers or dual prices pi, the Lagrangian function is defined as:

{ } { }( , ) 1 ( , ) ( , )

( , ) -N

ij ij i i ij jii j A i j i j A j j i A

L c x p s x x∈ = ∈ ∈

= + −

∑ ∑ ∑ ∑x p

Note that:

( , ) ( , )i ji j ij

j i i j

p x p x=∑ ∑

Therefore:

( , ) 1

( , )N

ij j i ij i ii j A i

L c p p x s p∈ =

= + − + ∑ ∑x p

The optimality conditions (b) specify that:

i j ijx j’

'j ix

is

75

*

( , ) 1

1 ( , )

( , ) min

min

N

ij j i ij i ii j A i

N

i i ij j i iji i j A

L c p p x s p

s p c p p x

≤ ≤∈ =

≤ ≤= ∈

= + − +

= + + −

∑ ∑

∑ ∑

0 x u

0 x u

x p

Instead of attempting to directly solve the original minimum cost network flow problem, the goal is to successively obtain updated dual price vectors p that solve the following dual problem:

max ( )hp

p

where the dual function can be decomposed into separable optimization problems over each arc:

1 ( , )

( ) ( )N

i i ij j ii i j A

h s p h p p= ∈

= + −∑ ∑p

with

( )( )

0( ) min

if

0 if

≤ ≤− = + −

+ − ≥ +=

< +

ij ijij i j ij j i ij

x u

ij j i ij i ij j

i ij j

h p p c p p x

c p p u p c p

p c p

Solution of the dual problem results in solution of the original minimum cost network flow problem. Notice that in the dual problem, the node mass balance constraints are temporarily relaxed since they are placed in the objective function via the Lagrangian function; hence, the term relaxation algorithm. The link capacity constraints remain explicitly accounted for. The objective is to find the optimal dual price vector p that will result in a solution that fully satisfies the node mass balance constraints. The advantage of this approach is that the inner minimization problem as defined by ( )ij i jh p p− is

extremely easy to solve, as seen above. Solution of the above separable optimization problems results in the following general arc optimality conditions associated with flow in arc (i,j) for a given dual price vector p:

arc: [ 0] if:

arc: [0 ] if:

arc: [ ] if:

ij i ij j

ij ij i ij j

ij ij i ij j

Inactive x p c p

Balanced x u p c p

Active x u p c p

= < +

≤ ≤ = +

= > +

A graphical representation of these arc optimality conditions is represented as the heavy line in this diagram. The optimal solution of the dual problem is found using a coordinate-wise dual ascent algorithm. Changes in the dual prices p are

0 uij

cij

Balanced

Active

Inactive

p - p

xij

i j

76

made along the directional derivative, where:

{ } { }

* *

( , ) ( , )

( ) , 1,...,ji ij i

j j i A j i j Ai

hx x s i N

p ∈ ∈

∂ = − + =∂ ∑ ∑p

and the directional derivative is:

1

1 1

( ) ( )'( ; ) ( ) N

S SN N

h p h py h

p d p d

∂ ∂ ∂ ∂= ∇ ⋅ = ⋅ + + ⋅∂ ∂ ∂ ∂

p pp d p d

Notice that evaluation of the directional derivative implies that the current flows are optimal. Since the arc optimality conditions specify that optimal flows are positive only for arcs that are active or balanced. Once a set of nodes i S∈ are identified for possible dual price changes, the directional derivative is calculated as:

( , ): active, , ( , ): active or balanced, ,

'( ; )S ji ij ij i j S i S i j i S j S i S

y u u s∉ ∈ ∈ ∉ ∈

= − +∑ ∑ ∑p d

where 1( ,..., )S Nd d=d , with

1 if

0 if i

i Sd

i S

∈= ∉

Notice in this formula that the directional derivative is not exactly evaluated, but rather is an approximation. However, it is a conservative approximation in that the directional derivative will be positive if a true ascent direction has been found. The first term only includes active arcs, where flows must be at the upper limits jiu in order for the arc

optimality conditions to be satisfied. The second term includes both active and balanced arcs, with the assumption made that flows in the balanced arcs are also at the upper bounds jiu . This latter assumption may not be correct, but notice that since these bounds

jiu are subtracted in the directional derivative calculation, then overestimation of these

flows would not indicate an ascent direction that was really a descent direction. That is, if '( ; ) 0Sy >p d , then direction Sd must be an ascent direction.

Define the surplus gi of node i as the difference between the total inflow into node i, less the total outflow from node i:

{ } { }( , ) ( , )i ji ij i

j j i A j i j A

g x x s∈ ∈

= − +∑ ∑

At the start of an iteration, an integer flow-node price pair (x, p) is assumed to be available which satisfy the arc optimality conditions, but not flow mass balance. The current iteration will indicate if: (i) the primal problem is infeasible (i.e., cannot find a node surplus gi > 0 for some i); (ii) (x, p) is optimal (i.e., gi = 0 for all i, implying that x is feasible and, since the arc optimality conditions are satisfied, is also optimal); or (iii) a new pair can be found that improves the dual objective function (i.e., gi > 0 for at least one node i). For the latter case, the iteration begins by selecting node k such that gk > 0. The iteration maintains the two sets: S and L ; where S L⊂ . At the initial iteration, set

{ }S = ∅ and { }L k= . A label is also maintained for all nodes , L which is an incoming

arc to that arc.

77

The goal is to maximize the dual objective function, which will result in solution of the original minimum cost network flow problem. A dual ascent direction is defined for node prices for those nodes contained in set S. Since set S usually contains a single node, the search procedure generally proceeds in one coordinate direction at a time of the node price vector p. Dual prices are changed in the dual ascent direction so as to increase the dual objective function. Since the goal is to eventually achieve a solution where all gi = 0 , a flow augmentation step occurs in the algorithm where a path through the network is defined from a node k where gk > 0 to a node j , where gj < 0. This means that flow can be increased along that path, resulting in improved node surplus conditions for both nodes. A.7 Typical Relaxation Iteration 0. Initialization

Initially start with flows 0 ( , )ijx i j A= ∀ ∈ and 0 , 1,...,ip i N= = . Select a

node k with node surplus gk > 0 [if no such node can be found, then the solution is optimal or infeasible]

{ } { }( , ) ( , )k jk kj k

j j k A j k j A

g x x s∈ ∈

= − +∑ ∑

• Let the set of labels { }L k=

• Let the direction vector set { }S = ∅

1. Choose a Node to Scan If: S L= (i.e., we are sure of an ascent direction);

GOTO Step 4 and perform price change Else: Select node i which is contained in the current set of labels, but not in

the current direction vector set; i.e., select i L S∈ − { }:S S i= ∪

GOTO Step 2 2. Label Neighboring Nodes of i • Check the directional derivative of the dual objective:

( , ): active, , ( , ): active or balanced, ,

'( ; )S ji ij ij i j S i S i j i S j S i S

y u u s∉ ∈ ∈ ∉ ∈

= − +∑ ∑ ∑p d

where direction vector dS = (d1,...,dN ), with 1 if

0 if i

i Sd

i S

∈= ∉

• If: ' 0y > , then current direction dS is an ascent direction; GOTO price change [Step 4]

Else: add to labeled set of neighboring nodes that can eventually result in identification of a flow augmentation path from node k to node j:

{ }:L L j= + for all nodes j such that:

C arc (j, i) is balanced and xji > 0 [assign label (j, i)], or C arc (i, j) is balanced and xij < uij [assign label (i, j)]

78

If: for every node j added to L, we have gj > 0, then we have not yet found a flow augmentation path: RETURN to Step 1

Else: Select one of the nodes j with: gj < 0; GOTO Step 3 3. Flow Augmentation

A flow augmentation path P has been determined to exist starting at node k and ending at the node j as found in Step 2. Since gk > 0 and gj < 0, then flow can be increased along the path such that gk will decrease towards zero, and gj will increase towards zero, subject to limitations.

Path P is constructed by tracing labels backward starting from j , where P+ is the set of all forward arcs and P - is the set of backward arcs:

Calculate:

C For all links in P+, ADD * to the current flows. C For all links in P-, SUBTRACT * from the current

flows min( ) ( , )

( , )

k

j

mn mn

mn

g

g

u x m n P

x m n P

δ +

−

−= − ∀ ∈ ∀ ∈

C GOTO next iteration 4. Price Change Set

balanced links ( , ) with ,

0 balanced links ( , ) with ,ij ij

ji

x u i j i S j S

x j i i S j S

= ∀ ∈ ∉

= ∀ ∈ ∉

Let

( ) , ,min

( ) 0, ,

ij i j ij ij

ji i j ji

c p p x u i S j S

c p p x i S j Sγ

− − < ∈ ∉ = − + − > ∈ ∉

Set if

otherwise i

ii

p i Sp

p

γ+ ∈=

GOTO Next Iteration

k j

Backward arcs: x > 0: set P-

Forward arcs: x < u: set P+

[i.e., can increase flow]

[i.e., can decrease flow]

79

A.8 Example Problem3 Consider the example network below, where exogenous flows are shown as supply and demand entering and leaving (respectively) each node. The link parameters are shown on each link, with all lower bounds set to zero. The objective is to find the minimum cost flow through the network that satisfies mass balance and all link flow upper bounds. We begin with an initial solution for the integer flow vector, dual price vector pair as (x, p) = (0, 0). Notice that this solution satisfies the arc optimality conditions, but violates feasibility since node surpluses 0ig ≠ .

3 Bertsekas, D., Linear and Network Optimization, The MIT Press, Cambridge, Massachusetts, 1991.

1

2

3

43 4

2

1

[uij , cij ]

[2,5] [1,2]

[2,1]

[3,4] [2,3]

[5,0]

80

ITERATION #1

Arc xij uij si sj cij pi pj gi gj State

(1,2) (1,3) (2,3) (2,4) (3,2) (3,4)

0 0 0 0 0 0

2 2 3 1 2 5

3 3 2 2 -1 -1

2 -1 -1 -4 2 -4

5 1 4 2 3 0

0 0 0 0 0 0

0 0 0 0 0 0

3 3 2 2 -1 -1

2 -1 -1 -4 2 -4

INACT INACT INACT INACT INACT BAL

Dual Objective Function = 0 + 0 = 0 Step 0. { }1L = ; { }S = ∅

1. Select i L S∈ − ; { }:S S i= ∪ ; so i = 1 and { }1S =

2. active active or balanced

' ji ij ii S

y u u s∈

= − +∑ ∑ ∑

= 0 - 0 + 3 > 0 [indicates that we can increase 1p ]

4 . No ijx adjustment is made at this iteration, since this is only done for

balanced arcs; calculate ij - cj ip p

[ ] 0 0 5min = 1 for arc(1,3)

0 0 1γ

− += − +

ii

[this assures we don’t go “too far”]

ITERATION #2

Arc xij uij si sj cij pi pj gi gj State

(1,2) (1,3) (2,3) (2,4) (3,2) (3,4)

0 0 0 0 0 0

2 2 3 1 2 5

3 3 2 2 -1 -1

2 -1 -1 -4 2 -4

5 1 4 2 3 0

1 1 0 0 0 0

0 0 0 0 0 0

3 3 2 2 -1 -1

2 -1 -1 -4 2 -4

INACT BAL INACT INACT INACT BAL

Dual Objective Function = 0 + 3 = 3 Step 0. { }1L = ; { }S = ∅

1. Select i L S∈ − ; { }1S =

2. y' = - 2 + 3 = 1 > 0 4. Arc (1,3) is balanced--

set 13x = 2

( = 0 - 1 + 5 = 4 (for arc (1,2) ); therefore, p1 = 1 + 4 = 5

0 uij

cij

pi - pj

xij

arc [1,3]

arc [1,2]

all other arcs

81

ITERATION #3

Arc xij uij si sj cij pi pj gi gj State

(1,2) (1,3) (2,3) (2,4) (3,2) (3,4)

0 2 0 0 0 0

2 2 3 1 2 5

3 3 2 2 -1 -1

2 -1 -1 -4 2 -4

5 1 4 2 3 0

5 5 0 0 0 0

0 0 0 0 0 0

1 1 2 2 1 1

2 1 1 -4 2 -4

BAL ACT INACT INACT INACT BAL

Dual Objective Function = 2 + 5 = 3⋅5 - 8 = 7 Step 0. L = {1}; 1g > 0 ; so { }S = ∅

[Note: node 1 is still selected, even though g2 is a greater--arbitrary] 1. S = {1} 2.

active or bal, ,

' 4 3 0ij ii S j S i S

y u s∈ ∉ ∈

= − + = − + <∑ ∑ [no improvement by increasing 1p ]

L: = L + {j} L = {1,2}: outflow link and balanced and ij ijx u<

Check if 2 0g > [yes!] [have not yet found flow augmentation path]

RETURN TO Step 1: 1. S = {1} ; L = {1,2} Select i 0 L - S ; i = 2 ; S = {1,2} 2. ' 2 5 0i

i S

y s∈

= = − + >∑ [we can increase 2p ]

4. γ = min {[ j ij ip c p+ − ] for arcs (2,3), (2,4)}

= min {4,2} = 2 Therefore, 1p = 5 +2 = 7; 2p = 0 +2 = 2 [for all nodes { }1,2i S∈ = ]

82

ITERATION #4

Arc xij uij si sj cij pi pj gi gj State

(1,2) (1,3) (2,3) (2,4) (3,2) (3,4)

0 2 0 0 0 0

2 2 3 1 2 5

3 3 2 2 -1 -1

2 -1 -1 -4 2 -4

5 1 4 2 3 0

7 7 2 2 0 0

2 0 0 0 2 0

1 1 2 2 1 1

2 1 1 -4 2 -4

BAL ACT INACT BAL INACT BAL

Dual Objective Function = -6⋅2 + 3⋅7 + 2⋅2 = 13 Step 0. L={1}; keep selecting node 1 since 1g > 0

1. S={1} 2. y' = (-2 -2) + 3 = -1< 0 L: = L + {j} L = {1,2} Check if 2g > 0 [Yes!]

RETURN to Step 1 1. S = {1,2}; L = {1,2} 2. y' = -2 - 1 + 5 = 2 > 0 4. Does ij ijx u= for all balanced arcs OUT?

Yes!--arc (2,4) Therefore, set 24x = 1

{ }min for arc (2,3) 2j ij ip c pγ = + − =

Therefore p1 = 7 + 2 = 9; p2 = 2 + 2 = 4

0 uij

cij

pi - pj

xij

arc (1,3)

arcs (1,2),(3,4),

arc (3,2)

(2,3),and (2,4)

83

ITERATION #5

Arc xij uij si sj cij pi pj gi gj State

(1,2) (1,3) (2,3) (2,4) (3,2) (3,4)

0 2 0 1 0 0

2 2 3 1 2 5

3 3 2 2 -1 -1

2 -1 -1 -4 2 -4

5 1 4 2 3 0

9 9 4 4 0 0

4 0 0 0 4 0

1 1 1 1 1 1

1 1 1 -3 1 -3

BAL ACT BAL ACT INACT BAL

Dual Objective Function = 2 + 2 + 9C1 + 4C1 = 17 Step 0. L={1}; g1 > 0 1. S={1} 2.

active or balanced

' ( 2 2) 3 0ij ii S

y u s∈

= − + = − − + <∑ ∑

L: = L + {j} with label (1,2) L = {1,2} Check if g2 > 0 ; Yes! RETURN to Step 1 1. S = {1,2}; L = {1,2} 2.

active/balanced OUT

' ( 2 3 1) 5 0ij ii S

y u s∈

= − + = − − − + <∑ ∑

L = L + {j} with label (2,3) L = {1,2, 3} Check if g3 > 0 ; Yes! RETURN to Step 1 1. S = {1,2}; L = {1,2, 3} Select i 0 L - S = 3 2.

active/balanced OUT

' ( 1 5) (3 2 1) 0ij ii S

y u s∈

= − + = − − + + − <∑ ∑

L: = L + {j} with label (3,4) L = {1,2,3,4} Check if g4 = < 0 ; Yes! = -3; GOTO Step 3: Flow Augmentation 3. Path of flow augmentation P is 1-2-3-4 [all forward arcs] ; so P+ = P

1

4

12 12

23 23

34 34

1 [ ]

( 4) [ ]

2 [ ] min 1

3 [ ]

5 [ ]

g

g

u x

u x

u x

δ

− − − −= = −

−

iiiii

1

2

3

4

+1

+1

+1

Flow IncreaseAlong Path

84

ITERATION #6

Arc xij uij cij pi pj gi gj State

(1,2) (1,3) (2,3) (2,4) (3,2) (3,4)

1 2 1 1 0 1

2 2 3 1 2 5

5 1 4 2 3 0

9 9 4 4 0 0

4 0 0 0 4 0

0 0 1 1 1 1

1 1 1 -2 1 -2

BAL ACT BAL ACT INACT BAL

Dual Objective Function = 5 + 2 + 4 + 2 + 4 = 17 Step 0. L = {2}; { }S = ∅ ; node k = 2

1. S = {2}; i = 2 2.

active/balanced OUT

' ( 3 1) 2 0ij ii S

y u s∈

= − + = − − + <∑ ∑

L = L + {j} ; add node 1 [label (1,2)] and node 3 [label (2,3)] L = {1,2, 3} ; check g1 = 0 and g3 = 1 [both > 0] RETURN to Step 1 1. set L - S = {1,3} Select node i = 3 Therefore: S = {2,3} 2. L = L + {j} ; add node 4 Check g4 = -2 < 0 ; GOTO Step 3: Flow Augmentation 3. Path P: 2-3-4 All forward arcs-- Therefore, P P+ =

y )' j

active INuji & j

active/balanced OUTuij % j

i0Ssi ' 2 & 5 % (2 % 1) ' &2 < 0

2

3

4+1

+1

Flow IncreaseAlong Path

2

4

34 34

23 23

1 [ ]

2[-g ] min =1

4 [ ]

2 [ ]

g

u x

u x

δ

= − −

iiii

85

ITERATION #7

Arc xij uij cij pi pj gi gj State

(1,2) (1,3) (2,3) (2,4) (3,2) (3,4)

1 2 2 1 0 2

2 2 3 1 2 5

5 1 4 2 3 0

9 9 4 4 0 0

4 0 0 0 4 0

0 0 0 0 1 1

0 1 1 -1 0 -1

BAL ACT BAL ACT INACT BAL

Dual Objective Function = 5 + 2 + 8 + 2 = 17 Step 0. L = {3} ; { }S = ∅ ; node k = 3

1. S = {3} ; node i=3 2.

active IN active/balanced OUT

' 2 5 1 4 0ji ij ii S

y u u s∈

= − + = + − − = − <∑ ∑ ∑

L: = L + {j} ; add node 2 [label (2,3)] and node 4 [label (3,4)] L = (2,3,4} Check g2 = 0 ; g4 = -1 [both < 0] 4. Path P+ = 3-4 ; node k = 3 ; node j = 4 * = min {1 [g3], 1 [-g4], 3 [u34 - x34]} = 1 FINAL SOLUTION

Arc xij uij cij pi pj gi gj State

(1,2) (1,3) (2,3) (2,4) (3,2) (3,4)

1 2 2 1 0 3

2 2 3 1 2 5

5 1 4 2 3 0

9 9 4 4 0 0

4 0 0 0 4 0

0 0 0 0 0 0

0 0 0 0 0 0

BAL ACT BAL ACT INACT BAL

Notice that 0ig = for all nodes. Therefore, dual objective = primal objective and all

complementary slackness conditions are satisfied. [primal: 5 + 2 + 8 + 2 = 17]

86

Appendix B STREAM-AQUIFER MODELING IN K-MODSIM

B.1 Analytical Equations The mathematical flow equation for general two dimensional flow in an unconfined groundwater aquifer can be derived from Darcy's Law and the principle of mass continuity. The resultant equation is a nonlinear, second-order partial differential equation known as the Boussinesq equation (Willis and Yeh, 1987):

x y

h h hK b K b Q S

x x y y t

∂ ∂ ∂ ∂ ∂ + + = ∂ ∂ ∂ ∂ ∂ (B.1)

where Kx, Ky are hydraulic conductivities along the x,y axes, respectively ( 1Lt− ); h is

potentiometric head (L); Q is net groundwater withdrawal per unit area ( 1Lt− ); S is storage coefficient; and t is time (t). Where variation in saturated thickness is small and the specific yield/storage coefficient is assumed constant, the governing groundwater equation can be written as a linear form of the Boussinesq equation:

2 2

2 2

h h hT Q S

tx y

∂ ∂ ∂+ + = ∂∂ ∂ (B.2)

where T is transmissivity = Kb ( 2 1L t− ), K is hydraulic conductivity ( 1Lt− ), and b is saturated thickness (L). Maddock (1974) showed that if the ratio of drawdown to saturated thickness is less than 20 percent, then for a nonlinear free-surface model (i.e., the Boussinesq equation), the linear contribution is between 75 to 100 percent of drawdown due to pumping. Accuracy of the linear model increases as the drawdown to saturated thickness ratio decreases. If the ratios are large, the Dupuit assumptions and the nonlinear flow equations are invalid. Since the governing groundwater equation is linear and time invariant, linear system theory can be applied via the principle of superposition (Bear, 1979). This principle states that the presence of one boundary condition does not affect the response produced by the presence of other boundary conditions and that there are no interactions among the responses produced by the various boundary conditions. It is then possible to analyze the effect of individual events and then linearly combine the results. Glover and Balmer (1954) and Glover (1968) presented an analytical procedure for determining depletion of flow in a nearby stream caused by pumping a well. Depletion flows were calculated using the distance of the well from the river, the properties of the

87

aquifer (i.e., storage coefficient and transmissivity), time of pumping and time from start of pumping. The following assumptions apply:

1. Aquifer is unconfined, homogeneous, isotropic, and of infinite extent 2. River is straight, fully penetrates the aquifer and is a constant head source. 3. Water table is initially horizontal and water is released instantaneously from

storage. 4. Well fully penetrates the aquifer. 5. Pumping is steady and drawdown is small compared to aquifer thickness. 6. Residual effects of previous pumping are negligible.

According to Glover (1968), the ratio of the rate of stream depletion to the rate of well discharge is:

14 /

s

w

Q aerf

Q tT S

= −

(B.3)

where Qs is rate of stream depletion ( 3 1L t− ); Qw is rate of well discharge ( 3 1L t− ); a is

perpendicular distance from well (L); t is pumping time ( t ); T is transmissivity ( 2 1L t− ); S is specific yield; and erf (z) is the error function. Glover (1977) extended the analytical approach to include bank storage, line source, return flows from irrigation, and intermittent well operation. Willis and Yeh (1987) presented a list of fifteen analytical response equations. Warner et al (1989) reviewed various analytical solutions to the artificial recharge problem, including Glover (1960), Hantush (1967), Rao and Sarma (1981), and Hunt (1971). The Hantush and Glover solutions were shown to be identical and were highly recommended for rectangular basins. It was also suggested that solutions for circular basins may be replaced by solutions for square basins with equivalent area. Madsen (1988) concluded that analytical models are not ideal for verifying the influence of existing wells on stream depletion, but are suitable as a tool for estimating impacts of new wells on streamflow depletion. Madsen (1988) also showed that analytical methods often overestimate stream depletion by failing to account for resistance near the stream. The major disadvantage of the analytical method is that nonpoint sources of flow are often approximated as point sources (Warner et al., 1986). Other limitations of analytical methods such as Glover's method include (Morel-Seytoux and Zhang, 1990): Method of averaging transmissivities over a heterogeneous aquifer is arbitrary; Procedure for calculating depletion from a certain reach (not the entire river) is

inconvenient, involving numerical integration, or inaccurate because of steady state assumptions;

In most cases, the river is not straight

88

Qazi and Danielson (1974) used a computer program based on the Glover equations to evaluate augmentation plans for wells, recharge lines, and pit operations in an alluvial aquifer. Contributory effects of only those pumped wells or recharge sources requiring evaluation are determined, and are independent of other interactions already in process such as: effects of precipitation, surface water application, evapotranspiration, or other wells, reservoirs, and ditches. Labadie, et al. (1983) used analytical solutions embedded in a conjunctive use model to consider groundwater pumping (Glover, 1977), reservoir seepage (Glover, 1977), canal seepage (McWhorter and Sunada, 1977), irrigation recharge (Maasland, 1959) and bank storage (Glover, 1977). Hantush and Marino (1989) developed a chance constrained stream-aquifer management model based on the Hantush (1959) analytical solution. Male and Mueller (1992) used the equations of Jenkins (1968) to develop a groundwater management model for prescribing groundwater use permits in Massachusetts. B.2 Discrete Kernel/Response Functions Most groundwater management scenarios require information only on select events in an aquifer. Extraneous information on drawdown and flow rates at noncritical locations is not only unnecessary but computationally prohibitive. Applying linear system theory to the groundwater equation allows the use of Green's function to solve the resulting non-homogeneous boundary value problem (Maddock, 1972). Response of the groundwater system due to external excitations such as pumping, recharge, or infiltration at any point in space and time can be expressed as a set of unit coefficients independent of the magnitude of the excitation. Integrated with a finite difference groundwater model, resultant flows can be superimposed to determine net effects at a single location due to a series of excitations or at a series of locations due to a single excitation. It is convenient to express the Boussinesq equation in terms of water table drawdown:

p

s s sT T Q S

x x y y t

∂ ∂ ∂ ∂ ∂ + + = ∂ ∂ ∂ ∂ ∂ (B.4)

where T aquifer transmissivity ( 2 1L t− ); s is water table drawdown (L); Qp is

groundwater withdrawal rate per unit area at well p ( 1Lt− ) ; S is storage coefficient; t is time (t); and x,y are horizontal coordinates (L). This equation can be solved using Green's function (Maddock, 1972):

0

( ) ( ) ( )t

w wp ps t t Q dδ τ τ τ= −∫ (B.5)

where sw(t) is drawdown at aquifer point w due to a single well pumping Qp at point p (L); and δwp is the kernel function (Green's function) of aquifer drawdown at point w due to a unit impulse excitation at p. The discrete form of the convolution equation for a

89

heterogeneous aquifer with finite boundaries is (Maddock, 1972; Morel-Seytoux and Daly, 1975):

1 1

( ) ( 1) ( )P t

w wp pp

s t t Qτ

δ τ τ= =

= − +∑∑ (B.6)

where t are now discrete time periods; sw(t) is drawdown from an initially horizontal (or initially steady) water table at any aquifer point w at the end of the period t (L); Qp(τ) is the mean pumping rate per unit area from well p during the period τ (pumped volume for the period) ( 1Lt− ); P is the total number of excitation points or wells; and δwp(t) is the discrete response or kernel coefficient representing the drawdown at the end of period t if a unit volume of water is withdrawn during the initial period from well p, with well pumping terminated indefinitely thereafter. Maddock (1972) first introduced the concept of a response function for a groundwater system, with drawdown in response to pumping stress modeled by a two-dimensional linear partial differential equation. This allowed an explicit coupling of a groundwater simulation model with a quadratic programming management model to optimize an economic objective of minimizing pumping costs subject to satisfying specified demands. Maddock (1974) used Green's function to extend this approach to the case of stream-aquifer interactions. Again, based on linear system theory and the Green's function, Morel-Seytoux and Daly (1975) developed a finite difference model to generate any aquifer response as an explicit function of pumping rates, which they referred to as a discrete kernel generator. The discrete kernel method has been utilized extensively as a tool for solving complex groundwater management problems (Morel-Seytoux, et al., 1981; Illangasekare and Morel-Seytoux, 1982; Illangasekare and Brannon, 1987; and Illangasekare and Morel-Seytoux, 1986). B.3 Parallel Drain Analogy for Stream-Aquifer Systems The interaction of a water table aquifer receiving recharge from irrigation and precipitation, and an interconnected stream, can be modeled utilizing the method developed by Maasland (1959). This method was developed for a parallel drain system and can be applied to a stream-aquifer system as well. The idealized parallel drain system is shown in Fig. B.1.

90

The nonlinear partial differential equation for one-dimensional groundwater flow is

( ) h hK d h S

x x t

∂ ∂ ∂+ =∂ ∂ ∂

(B.7)

where K is permeability of the aquifer ( 1Lt− ); d is original saturated thickness (L); S is specific yield; h is height of the water table measured from the assumed original stable water table level (L); d is depth measured from the assumed original stable water table level to the impermeable boundary (L); x is distance measured along the path of flow (L); and t is time. By assuming h is small compared to d, the linearized form of Eq. B.7 is:

2

2

h h

txα ∂ ∂=

∂∂ (B.8)

where T

Sα = ; T is transmissivity, which equals K⋅ d; and the boundary conditions are:

h = 0 when x = 0 for t > 0 h = 0 when x = L for t > 0 h = H when t = 0 for 0 < x < L Maasland (1959) obtained the solution as:

2 2

21,3,5...

4 1exp sin

n

H n t n xh

n LL

π α ππ

∞

=

− =

∑ (B.9)

Drain

h

d

H

Fig. B.1 Parallel drain analogy for stream-aquifer systems.

Drain

L

Water Table

Ground Surface

Barrier

x K S

91

where H is initial uniform height of recharge water from an initial event and L is spacing of the parallel drains. The volume per unit length of water remaining to be drained (L2) is

0

L

dV S h dx= ∫ (B.10)

and the fraction remaining to be drained is

dVF

V= (B.11)

where initial drainable volume is V S H L= ⋅ ⋅ (B.12) Therefore

0

L

S h dx

FS H L

=⋅ ⋅

∫ (B.13)

Assuming H = 1 represents a unit initial recharge event, substitution of h from Eq. B.9, and integration results in:

2 22 2 2

1,3,5...

8 1exp

n

tF n

n L

αππ

∞

=

= −

∑ (B.14)

representing the fraction of the total initially drainable volume due to the unit recharge event that is in the aquifer at the end of time t and available for flow to the drains. For any time t from the beginning of the recharge event, F can be predetermined. The difference of successive F values over two adjacent discrete time periods represents the flow fraction to the drains during that time interval.

( )

2 21 2 2 2

1,3,5...

2 22 2 2

1,3,5...

8 1exp

18 1 exp

k kn

n

k tF F n

n L

k tn

n L

αππ

απ

π

∞

−=

∞

=

∆ − = −

− ∆− −

∑

∑ (B.15)

Define

1k k kF Fδ −= − (B.16)

92

where δk is a unit response coefficient or discrete kernel for a unit recharge rate at initial time interval k = 1. Note that the recharge rate is idealized as being uniformly distributed over the width of spacing between drains. B.4 Return Flow Calculations Consider the idealized stream-aquifer system as shown in Fig. B.2. The river is assumed to be located at the center of the valley. The solution described above can be applied directly with L equal to the valley width. The analogy is applicable since the middle section of the parallel drains is a no-flow boundary and is analogous to either the left boundary or the right boundary of the stream-aquifer system. If the parallel drain system is divided in half at the no flow boundary and rearranged to bring the drains into coincidence, the direct analogy with the stream-aquifer system is evident. The drains are replaced by the river and the flow to the drains represents return flow to the river.

For cases where the river is not located at the center of the valley, the above solution (Eq. B.14) is still applicable with L equal to twice the width W of either side of the valley (i.e., L2 = 4W2. Fraction F can be determined for each side of the valley and return flows computed separately. Again, it is idealized that recharge events are uniformly distributed over each side of the valley. Let N be the total number of time intervals of length ∆t and Ik the recharge rate during the k-th time interval, where k < N , as shown in Fig. B.3. For any demand node i and current time period k, the total return flow IRFik from the current and previous time periods due to groundwater recharge is calculated using linear superposition:

, 1 , 11

; = 0 for 1k

ik i i k i kIRF I k Nτ τ ττ

δ δ τ− + − +=

= ⋅ − + >∑ (B.17)

L

Water Table River

Ground Surface

Barrier

x

d

h

K S

Fig. B.2. Idealized stream-aquifer system (Glover, 1977).

93

where ikIRF is the infiltration rate at node i, period k , and , 1i k τδ − + is the response or

discrete kernel coefficient defined for node i, period k - τ+1. In K-MODSIM, upper bounds on return flow links (Fig. 3) are adjusted iteratively as follows: (1) all upper bounds are first set equal to the return flows computed from previous activities; (2) K-MODSIM is next run for the current period using these bounds, with return flows from all sources recomputed using available link flows obtained from this solution; (3) if current return flows agree with previous estimates, the process terminates; otherwise, return to step 2 and repeat until convergence is achieved. B.5 Stream Depletion from Pumping

The same approach used for calculating return flows is also applied to calculation of stream depletion due to pumping PSDik , where

, 1 , 11

; = 0 for 1k

ik i i k i kPSD P k Nτ τ ττ

α α τ− + − +=

= ⋅ − + >∑ (B.18)

In the case of groundwater withdrawal iPτ , the same principles described above are

applicable to determining response coefficient kernels , 1i k τα − + , but for river depletion

rather than return flows. Again, it is idealized that pumping withdrawals are uniformly distributed over each side of the valley, rather than attempting to model individual wells. Since the computation is sequentially carried out period by period in K-MODSIM, the current period stream-aquifer interactions are contingent upon stresses during previous periods. Therefore, it is recommended to run K-MODSIM for an initial N periods for start-up or initialization purposes, such that after N periods, the model output can be trusted to properly account for past history. Specification of N is left to the user.

Rec

har

ge

Rat

e I

k

Time Interval k

1 2 3 . . . k-2 k-1 k . . . N-2 N-1 N

Fig. B.3 Series of recharge events

94

B.6 Canal Seepage Seepage from a canal or a stream is assumed to correspond to a line source of recharge water. For a one-dimensional line source in an infinite aquifer, as shown in Figure B.4, the governing flow equation is (McWhorter and Sunada, 1977):

2

2

q q

txα ∂ ∂=

∂∂ (B.19)

where x is the Cartesian coordinate in the horizontal plane (L) and q is the flow rate or Darcy velocity ( 2 1L t− ), calculated as:

h

q Kt

∂= −∂

(B.20)

The solution is (McWhorter and Sunada, 1977):

x

erfc2 4 t

Iq

α =

(B.21)

where I is the one dimensional magnitude of the source ( 1Lt− ), with erfc(z) representing the complementary error function:

( ) 22erfc u

z

z e duπ

∞−= ∫ (B.22)

assuming the following boundary and initial conditions:

at 020 as

0 as 0; for all

Iq x

q x

q t x

= =

= →∞= =

(B.23)

x

Water Table

Ground Surface

Barrier

K S

h

q

Ck

d

Fig. B.4. Illustration of line source for canal seepage.

95

Define 0 2

Iq = as the applied line source flow rate in the aquifer at the line source

location. Note that the denominator of two is necessary since q flows in two horizontal directions. Integrating Eq. B.20 from zero to t results in the ratio of the volume of return flow to the stream and the line source volume of seepage applied up to time t:

2 2

0

21 erfc exp

2 44 4

v x x x x

q t t tt tα αα α π = + − − ⋅

(B.24)

This solution is for a continuous application of a line source. After termination of the source, the residual effect still contributes flow to the stream. The residual is taken into account by assuming an imaginary pumping source at the same location and initiating pumpage at the same rate as the recharge source from the time recharge terminates. The volume ratio at any time after recharge ceases is the difference between the volume ratio obtained if recharge had continued and the volume ratio obtained from pumping of the imaginary pumping source. For a discrete time interval, if the applied line source volume equals one, the volume ratio is in essence the unit response of line source or canal seepage. Let φ represent the unit response of canal seepage. Then for canal link ℓ, the total return flow CRFℓk from canal seepage Cℓ1,,...,Cℓk during each time interval k is:

, 1 , 11

; = 0 for 1k

k k kCRF C k Nτ τ ττ

φ φ τ− + − +=

= ⋅ − + >∑ (B.25)

B.7 Point Source Water Application Reservoir seepage RSik is defined as a point source application for storage node i , time period k. The effect on the stream corresponds to the effect of a recharge well, which in turn has the same absolute flow magnitude as a pumping well, with the flow direction reversed. This solution turns out to be exactly the same as that for the line source solution (Glover, 1977). Therefore, Cℓτ is replaced with RSiτ in Eq. B.24, with the resulting return flow defined as RRFik. Again, there is little error in assuming reservoir seepage as a point source, as long as the reservoir surface area is small in comparison with the area of the subsystem containing it. For reservoir i during time period k, the total return flow RRFik from reservoir seepage, based on current and previous period seepage, is

, 1 , 11

; = 0 for 1k

ik i i k i kRRF RS k Nτ τ ττ

φ φ τ− + − +=

= ⋅ − + >∑ (B.26)

96

B.8. Stream Depletion Factor Method (sdf) Jenkins (1968) solved the Glover equation graphically by developing dimensionless curves and tables to compute the rate and volume of stream depletion by wells. The stream depletion factor (sdf) was arbitrarily chosen as time in days where the volume of stream depletion is 28% of the volume pumped during time t, and can be expressed as:

2a S

sdfT

= (B.27)

where a is perpendicular distance from the pumped well to the stream (L); S is specific yield of the aquifer (dimensionless); and T is transmissivity ( 2 1L t− ). In a complex system, the value of sdf at any location depends on the integrated effects of irregular impermeable boundaries, stream meanders, aquifer properties, areal variation, distance from the stream, and hydraulic connection between stream and aquifer. The basic assumptions are similar to those associated with the Glover equation: Moulder and Jenkins (1969) introduced the sdf concept to a digital model and the USGS used it to generate groundwater response coefficients for developing regional models (Taylor and Luckey, 1972; Hurr, 1974; Hurr and Burns, 1980; and Warner et al., 1986) and groundwater sdf contour maps (Hurr, et al., 1972). B.9. Response Functions from Finite Difference Groundwater Models The Boussinesq partial differential equation for groundwater flow in a heterogeneous and anisotropic medium (Eq. B.1) can be solved using finite difference or finite element numerical approximation methods (Willis and Yeh, 1987). Finite difference methods require that a regular Cartesian grid be defined over the domain of the region to be modeled. The groundwater basin is spatially discretized into a grid structure represented by a finite number of rectangular cells. Differential terms in the partial differential equation are replaced with numerical approximations calculated from differences in potentiometric head at the cell locations. All aquifer parameters, heads and hydrologic data are assumed to be constant and homogeneously distributed within a grid cell. This numerical approximation results in replacement of the original partial differential equation with a system of simultaneous linear difference equations. The most popular finite difference groundwater modeling system is MODFLOW, developed by the U.S. Geological Survey (Harbaugh, et al., 2000). The finite element method is another approach to numerical modeling of groundwater basins, but is less popular and generally more computationally time consuming than the finite difference method, although it offers the advantage of more accurate representation of irregular aquifer boundaries. Application of numerical models such as MODFLOW allow relaxation of most of the idealized assumptions associated with analytical modeling approaches such as the Glover method. Hartwell (1987) compared results from a model based on the Glover solution, the sdf method, and a finite difference model for a recharge site along the along the South Platte River, Colorado. Use of the finite difference model was recommended in this

97

study since it provided more accurate return flow calculations than the other methods. Sophocleous et al. (1995) also compared the Glover analytical solution with MODFLOW, concluding that the latter was preferred for more accurately treating irregular boundary conditions, streambed aquifer hydraulic conductivity, partial stream penetration into the aquifer and heterogeneity of the aquifer porous media. Numerical groundwater flow models such as MODFLOW provide more accurate modeling of stream-aquifer systems, but also require extensive field data collection and observation well records for model calibration and verification. Once calibrated, attempting to link MODFLOW with K-MODSIM would be computationally intractable. However, use of the linear form of the Boussinesq equation (Eq. B.2) allows application of Green’s function to solve the resulting nonhomogeneous boundary value problem (Maddock, 1972). Response of the finite difference groundwater modeling system due to external excitations such as pumping, recharge, or infiltration at any point in space and time can be expressed as a set of unit response coefficients independent of the magnitude of the excitation. These response coefficients are similar in concept with those calculated via the Glover model in Eq. B.16, but incorporate more realistic assumptions associated with the MODFLOW model. Although the linear form of the Boussinesq equation is only valid for confined aquifers where transmissivity does not vary with head, it can be applied to unconfined aquifers with reasonable accuracy as long as ∆h/∆H < 0.10, where ∆h is the change in groundwater elevation and H is saturated thickness of the aquifer. Although a calibrated MODFLOW model can be applied to generating response functions for stream-aquifer interactions, Maddock and Lacher (1991) developed MODRSP as a variation of MODFLOW designed specifically for calculating response or kernel functions for stream-aquifer interactions. Transient, spatially distributed stream-aquifer response coefficients are automatically generated using MODRSP for allocating groundwater return/depletion flows to multiple return/depletion flow grid cell locations. MODRSP calculates responses for one well or recharge site at a time over the total simulation period, assuming a unit stress has been applied during the first period and discontinued for the remainder of the simulation. Response functions calculated in this way incorporate all of the complex characteristics of the stream-aquifer system into a unique cause-effect relationship. Since MODRSP is a modification of the USGS MODFLOW finite difference groundwater model, it uses many of the same input data and file structures as MODFLOW. This means that MODFLOW can initially be applied and calibrated for the study area, allowing application of powerful packages such as GMS (The Department of Defense Groundwater Modeling System) (EMRL, 2005) that provide powerful geographic information system (GIS) tools for preparation of MODFLOW data sets. The MODFLOW input data sets developed from application of GMS can, for the most part, be directly utilized in MODRSP, although MODFLOW must be applied in conjunction with the constant transmissivity assumption associated with the linear form of the Boussinesq equation. Additional assumptions that must be adhered to in applying MODFLOW for application of MODRSP include:

98

The MODRSP river package does not require data on river stage height and the head at the bottom of the streambed. For return flow/depletion responses in river reaches, MODRSP assumes uniform water levels over each stream reach that are constant over each stress period. Flow conditions in the stream are assumed to vary insignificantly during stress periods. If streams go dry or overflow their banks during a stress period, it is assumed such events are of short duration and have negligible effect on long-term stream-aquifer interaction.

Since all starting heads are set to zero in MODRSP, a starting head input file is not required.

Since MODRSP is a linear model, transmissivity and storage coefficients are considered constant and must be entered as input data.

It is unnecessary to prepare a well package since pumping data are not read into MODRSP.

Response coefficient output data generated by MODRSP can be formatted to include well/recharge grid location, response grid location, stress period, and calculated response coefficient for that period. Typical database structure for response coefficient output data is presented in Table 2.

The following modifications were made to MODRSP by Fredericks and Labadie (1995): The program was modified to allow dynamic array dimensioning up to available

RAM memory. The modules RRIV.FOR and RPGM.FOR source code were modified to reduce

unnecessary output to a river response file. In line 1 of the RRIV input file, field 41 to 50, a decimal value for the variable, RDROP, can be input. Response coefficients

Table 2. River Capture Response Functions from MODRSP RIVER CAPTURE RESPONSE FUNCTIONS ________________________________ RIVER PUMP REACH WELL TIME # K I J # K I J PER RF [0] _______________________________________________________ 1 1 2 2 1 1 5 10 1 .1644417E-02 2 1 2 3 1 1 5 10 1 .2860665E-02 3 1 2 4 1 1 5 10 1 .3791862E-02 4 1 3 4 1 1 5 10 1 .4539182E-02 5 1 4 4 1 1 5 10 1 .5317831E-02 6 1 4 5 1 1 5 10 1 .6792709E-02 7 1 4 6 1 1 5 10 1 .8866128E-02 8 1 4 7 1 1 5 10 1 .1095925E-01 9 1 4 8 1 1 5 10 1 .1415163E-01 10 1 4 9 1 1 5 10 1 .1891528E-01 11 1 4 10 1 1 5 10 1 .2429373E-01 12 1 4 11 1 1 5 10 1 .1800508E-01 13 1 3 11 1 1 5 10 1 .1408879E-01 14 1 3 12 1 1 5 10 1 .1173625E-01 15 1 3 13 1 1 5 10 1 .9830805E-02

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lower than this value will not be printed to the river response output file, thereby reducing the size of the river response output file by eliminating zero value response functions.

The modules RRIV.FOR and RPGM.FOR were modified to terminate a processing loop for a specific well when the calculated response coefficient values fall below a specified lower limit.

The modules RRIV.FOR and RPGM.FOR were modified to read in a river reach file that assigns a specific river reach value to each river reach grid cell and then sums the response coefficients by river reach.

The module RPGM.FOR was modified to read in a recharge site file that assigns a recharge site number to each well grid cell number.

An example finite difference grid for modeling a portion of the South Platte River basin in Colorado is shown Fig. B.5 (Fredericks, et al., 1998). For this case study, two different sets of response coefficients were generated: numerical coefficients calculated using the MODRSP finite difference groundwater model and analytical coefficients calculated with the Glover equation using predefined sdf values. The

Aquifer Boundary

South Platte River

0 0

140

370

DENVER Scale in Miles

0 10 20 30

Finite-Difference Grid 140 x 370 cells

(1000 ft x 1000 ft)

N

Fig. B.5 MODRSP finite difference grid for numerical groundwater model.

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response functions for a particular river reach in the study area are compared in Fig. B.6. It was found that use of the analytically-based sdf coefficients produces significantly lower net river return flow values when compared with coefficients derived from the numerically based finite difference model.

Fig. B.6 Comparison of MODSRP and sdf computed response functions.

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