Cost analysis of water and sediment diversions to optimize landbuilding in the Mississippi River delta

Melissa A. Kenney,1,2 Benjamin F. Hobbs,1,2 David Mohrig,2,3 Hongtai Huang,1 Jeffrey A. Nittrouer,2,4

Wonsuck Kim,3 and Gary Parker2,5

Received 10 May 2012; revised 9 February 2013; accepted 11 February 2013.

[1] Land loss in the Mississippi River delta caused by subsidence and erosion has resultedin habitat loss and increased exposure of settled areas to storm surge risks. There is debateover the most cost-efficient and geomorphologically feasible projects to build land by riverdiversions, namely, whether a larger number of small, or a lesser number of large,engineered diversions provide the most efficient outcomes. This study uses an optimizationframework to identify portfolios of diversions that are efficient for three general restorationobjectives: maximize land built, minimize cost, and minimize water diverted. Theframework links the following models: (1) a hydraulic water and sediment diversion modelthat, for a given structural design for a diversion, estimates the volume of water andsediment diverted; (2) a geomorphological land-building model that estimates the amountof land built over a time period, given the volume of water and sediment; and (3) astatistical model of investment cost as a function of diversion depth and width. An efficientportfolio is found by optimizing one objective subject to constraints on achievement of theother two; then by permuting those constraints, we find distinct portfolios that representtrade-offs among the objectives. Although the analysis explores generic relationshipsamong size, cost, and land building (and thus does not consider specific project proposals orlocations), the results demonstrate that large-scale land building (>200 km2) programs thatoperate over a time span of 50 years require deep diversions because of the enhancedefficiency of sand extraction per unit water. This conclusion applies whether or not there aresignificant scale economies or diseconomies associated with wider and deeper diversions.

Citation: Kenney, M. A., B. F. Hobbs, D. Mohrig, H. Huang, J. A. Nittrouer, W. Kim, and G. Parker (2013), Cost analysis of waterand sediment diversions to optimize land building in the Mississippi River delta, Water Resour. Res., 49, doi:10.1002/wrcr.20139.

1. Introduction

[2] The Mississippi River delta has lost over 1.2 millionacres (4900 km2) of land since the 1930s [Walker et al.,1987; Coastal Louisiana Ecosystem Assessment and Resto-ration, 2006; Coastal Protection and Restoration Authorityof Louisiana (CPRA), 2007; Morton et al., 2005a; Mortonand Bernier, 2010]. Over the next 50 years, State of Louisi-ana [2012a] predicts that the additional amount of land lostif no action is taken could be as much as 1900 km2. It is

generally accepted that the historic loss was primarilycaused by sea level rise, subsidence resulting from oil andgas production, and sediment starvation due to levee con-finement of the Mississippi River as well as construction ofthe Missouri River dams [Blum and Roberts, 2009; Kolkeret al., 2011]. Prior to dam and levee building and oil andgas exploitation in the twentieth century, long-term rates ofland building roughly balanced land loss [Frazier, 1967;Penland et al., 1988; Paola et al., 2011]. Now, however,levees confine water and sediment to the main channel,thereby preventing river water from flowing onto the adja-cent wetland environments. Consequently, deposition ofsediment is dramatically lower in shallow areas of the delta[Baumann et al., 1984; Walker et al., 1987] and cannotkeep pace with wetland lowering due to subsidence[Morton et al., 2005a]. Sediment in the river channel isthus transported all the way to Venice, Louisiana, theapproximate beginning of the river’s multichannel outlet tothe Gulf of Mexico with little deposition in the adjacentwetland environments [Walker et al., 1987; Boesch et al.,1994; Day et al., 2000]. There is a need for science toinform coastal restoration decisions given the range ofstressors and desired benefits from restoring a dynamicinteraction between the Mississippi River and its coasts[Day et al., 2007; Groves et al., 2012].

1Department of Geography and Environmental Engineering, Johns Hop-kins University, Baltimore, Maryland, USA.

2National Center for Earth-Surface Dynamics, University of Minnesota,Minneapolis, Minnesota, USA.

3Department of Geological Sciences, Jackson School of Geosciences,University of Texas, Austin, Texas, USA.

4Department of Earth Science, Rice University, Houston, Texas, USA.5Department of Civil and Environmental Engineering and Department

of Geology, University of Illinois at Urbana-Champaign, Urbana, Illinois,USA.

Corresponding author: M. A. Kenney, Earth System Science Interdisci-plinary Center, University of Maryland, College Park, MD 20740, USA.(kenney@umd.edu)

©2013. American Geophysical Union. All Rights Reserved.0043-1397/13/10.1002/wrcr.20139

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WATER RESOURCES RESEARCH, VOL. 49, 1–18, doi:10.1002/wrcr.20139, 2013

[3] Land loss in southern Louisiana has degradedhabitat, interfered with human activities, and increasedexposure of New Orleans and other settled areas to stormsurge risks [U.S. Army Corps of Engineers, 1963; Boeschet al., 1994; Tidwell, 2003; Fritz et al., 2007; Lowe, 2008;National Research Council, 2009; Bunya et al., 2010]. Tomitigate these impacts, there have been several delta resto-ration planning efforts over the past several decades thathave considered a range of environmental and socialobjectives. Notably, the Coastal Wetlands Planning,Protection and Restoration Act provides funds for coastalwetland restoration projects (http://lacoast.gov/new/Projects/List.aspx). Those projects have many goals, withland building being just one of them, and have largelyfocused on projects that are small relative to the projectscale that we explore here. It is possible that environmentaland social concerns will constrain the size of individualprojects or require them to be sited in areas in which landbuilding is less effective. Nevertheless, for broad ecologicalbenefits, a focus on land building is appropriate given thatmany coastal ecosystem services and natural coastalprotection methods [Twilley, 2007; Costanza et al., 2008]are strongly and positively correlated with the amountof land [Coastal Louisiana Ecosystem Assessment andRestoration, 2006].

[4] Despite the relatively recent interest in land-buildingdiversions in the Mississippi delta, river diversions ondeltaic land have been primarily built to reduce the risk oflevee overtopping during floods. In particular, this wasprecisely the purpose of the Bonnet Carr�e (http://www.mvn.usace.army.mil/recreation/rec_bonnetcarre.asp)and Morganza spillways upstream of New Orleans [Davis,1993; Lane et al., 2001; U.S. Army Corps of Engineers,2009; Nittrouer et al., 2012]. Diversions designed to buildland are rarer, and there are no diversions whose purpose islarge-scale land building (say, >100 km2 over 50 years).Some small diversions, such as Caernarvon, have been con-structed on the lower Mississippi River for purposes otherthan land building (e.g., freshwater diversion) [Lane et al.,1999]. Coastal restoration strategies and projects have beendiscussed in the literature [Boesch et al., 1994; Allison andMeselhe, 2010] and are being considered by Louisiana andthe U.S. government [National Research Council, 2006;U.S. Army Corps of Engineers, 2008; State of Louisiana,2012b]. Although the delta’s original footprint cannot berestored, there is likely sufficient sediment in the Missis-sippi/Atchafalaya system to counteract most or all futureland loss if we understand where and how to use the avail-able sediment.

[5] While the State of Louisiana [2012a] suggests that theamount of land loss in the next 50 years could be as large as1900 km2, recent data assembled by Morton et al. [2005a]and Morton and Bernier [2010] indicate that this value couldbe an overestimate; a better value for the amount of landloss is closer to 1350 km2 over the next 50 years, based onthe measurements of loss since 1978. A recent analysis byKim et al. [2009b] uses a numerical model to predict thatabout 600 km2 of land could be developed in 50 years bydiverting less than half of the total Mississippi River sandload in two large-scale diversions. Thus, by extracting just afraction of the total water and sediment load of the river, asignificant portion of the estimated land loss could be offset.

[6] The water diverted from the Mississippi River in theKim et al. [2009b] analysis is 11,500–18,000 m3/s duringflood conditions, well below the 21,800 m3/s limit imposedin the Louisiana Coastal Master Plan [State of Louisiana,2012a, Appendix B]. Furthermore, considering contribu-tions from the fine-grain suspended sediment load (i.e.,mud fraction) and land built from organic sediment wouldfurther boost estimates for land growth via controlleddiversions [Wright and Parker, 2005a, 2005b; Kim et al.,2009b]. For example, core analyses by Wilson and Allison[2008] show that the fraction of the aggrading land surfacecomposed of organic materials averages 29%, based on thesediment cores collected from the wetlands of the Missis-sippi delta (Breton Sound and Barataria Bay). This estimateis close to that by Morton et al. [2005b], who measure thevolume of organics in cores recovered throughout the Mis-sissippi delta as about 27%. Thus, water and sedimentdiversions have the potential to build significant amountsof land. Combining predicted land building from diversionson the modern Mississippi River with ongoing land growthin the Atchafalaya delta and Wax Lake Delta (approxi-mately 5 km2/yr) [Paola et al., 2011] suggests that there isthe potential for land growth in the entire Mississippi Riverdelta system that could at the very least match, if not out-pace, predicted future land loss.

[7] Given that large-scale diversions could cost billionsof dollars, it is important to identify alternative portfoliosof diversion projects that cost-effectively meet land goalswhile satisfying environmental constraints. An efficient, or‘‘nondominated’’/‘‘pareto optimal’’ [Cohon and Marks,1975], alternative is defined as one for which there is noother option that performs as well in terms of cost, land cre-ation, and perhaps other objectives while being strictly bet-ter in at least one of those objectives. If there exists someother such alternative, then the first alternative is said to beinefficient or dominated by the second. Here, we define al-ternative portfolios that are efficient in terms of three majorobjectives: maximize land built, minimize cost, and mini-mize water diverted. These are not the only objectives thatwould interest stakeholders ; however, these three objec-tives are among the most important in coastal restorationdiversion decisions. In particular, the master plan uses areaof land built as a proxy for ecological benefits of wetlandrestoration and water diverted as a surrogate for possibleimpacts on Mississippi river navigation and water supply[State of Louisiana, 2012a]. Though other objectives areimportant for site-specific planning decisions, such consid-erations are outside of the scope of this study because ourpurpose is to provide a generic analysis of the effect ofscale economies upon the relative desirability of largeversus small diversions.

[8] Economic analysis has previously been applied tocoastal wetland restoration in the Mississippi delta andelsewhere. However, most of those analyses have empha-sized quantifying benefits [Costanza et al., 2008] or theoverall balance of benefits and costs of restoration[Spurgeon, 1999], rather than comparing the cost-effective-ness of various project sizes. The exception is Turner andBoyer [1997], who argued that there is a greater return oninvestment (in terms of land built per unit cost) for smallerdiversion projects. However, since that paper was pub-lished, there have been significant improvements in our

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ability to quantitatively estimate volume of new land builtthrough controlled diversions [Kim et al., 2009b; Paola etal., 2011]. Because of the expense of land-building pro-posals under consideration, it is worth revisiting the ques-tion of efficient diversion sizes for building land.

[9] This study considers these trade-offs within a generalanalysis of scale economies and diseconomies in landbuilding from diversions in the lower Mississippi Riverdelta. The analysis addresses the question: how many proj-ects of what size should be built in order to maximize theamount of land built by a particular time (50 years in thefuture)? The purpose is to gain general insights on the effi-cient scale of diversion projects, considering trade-offsamong costs, land built, and amount of water diverted; andto understand generally how assumptions about scaleeconomies of project construction affect those conclusions.

[10] The analysis of diversions presented here involvesdevelopment of a set of linked engineering, geomorphol-ogy, and economic models, using the most current under-standing, that are integrated using optimization.Uncertainty in the cost, hydraulic, and geomorphologicalrelationships could imply that an adaptive planning processwould be beneficial ; analysis of uncertainties and the valueof learning are left for future research. The analysis is gen-eral in nature, and thus, it is not of sufficient detail for site-specific project recommendations. After describing theframework and each of the three models, we present theresults of the analysis for a single diversion, a portfolio ofdiversions, and sensitivity analyses of key assumptions andfinally discuss some implications of this work.

2. Methods

[11] Identification of the optimal diversion design tomaximize land building, for either a single project or port-folio of projects, relies on the coupling of engineering, geo-morphology, and economics models within an optimizationframework. Specifically, we develop a water and sedimentdiversion model that for given box-culvert design elementspredicts the amount of water and sediment transportedthrough the diversion. Then given the water and sedimentdelivery for a specified diversion, this information can beused in a land-building model to predict the amount of landbuilt in a given time period. We also estimate a diversioncost model as a function of the diversion depth and widthbased on existing diversion structures. Finally, we integratethe land building and cost models to identify portfolios ofdiversions that are efficient in terms of the three objectives.Later, we first present the optimization framework that inte-grates the outputs from the diversion, land building, andcost models; each model is then discussed in separatesubsections.

2.1. Optimization Framework

[12] The purpose of the model is to identify efficientportfolios of diversions in terms of the objectives of maxi-mizing land built within 50 years, minimizing project cost,and minimizing the amount of water that is diverted. Aportfolio is defined as a collection of projects of possiblydifferent designs whose aggregate cost, land built, andwater diverted is assumed to be the sum of the correspond-ing cost, land built, and water diverted by each constituent

project. That is, for simplicity we disregard possible inter-actions among the diversion projects in multiple places, forinstance the effect of upstream diversions on sedimentavailable downstream. In this way, we reduce the complex-ity of feedbacks among diversions, enabling us to gaininsights concerning generic questions of optimal projectscale, which we obtain by coupling the models within thisframework. Because interactions might be significant if alarge fraction of sand and water in the Mississippi River isdiverted by multiple projects, actual decisions about loca-tion and sizing should consider these interactions.

[13] The projects are assumed to divert water and sedi-ment from the Mississippi River only during the yearlyhigh flow, which we take here to be the duration for a rep-resentative flood period (2.5 months). Minimizing totalwater diverted is assumed to be an objective in projectplanning; as mentioned earlier, this is consistent with themaster plan, which stated explicitly that ‘‘due to naviga-tion, municipal water and other uses of the MississippiRiver, a maximum discharge capacity from sedimentdiversions of 21,804 m3/s (770,000 cfs) was used toensure an adequate flow of water in the Mississippi Riverat all times’’ [State of Louisiana, 2012a, Appendix B]. Inour multiobjective trade-off analysis, we considered arange of water diversion limitations of up to 25,000 m3/s,which is roughly two thirds of a major flood flow for theMississippi River downstream of the Old River ControlStructure. The assumption of a limit on water availabilityfor diversions is also consistent with assumptions made inprevious analyses concerning diversion magnitudes thatwould avoid unfavorable impacts on navigation, which inKim et al. [2009b] were set at a maximum of 45% of theannual flow. However, it may be possible to relax this li-mitation by redesigning navigation channels (e.g., flood-gates and armored banklines) to allow rerouting ofshipping traffic. Thus, our solutions consider limits upondiverted flow as high as 25,000 m3/s.

[14] To generate efficient solutions, we adopt the nowclassic constraint method of multiple objective program-ming, in which all objectives except one are constrained,and the remaining one is optimized [Cohon and Marks,1975]. Then by tightening or loosening the constraints, dif-ferent solutions in the efficient set of alternatives can begenerated. Here, we choose cost as the objective to be opti-mized; the choice is arbitrary and does not affect the effi-cient set.

[15] The resulting optimization model can be stated suc-cinctly as follows: choose the number (ni) of projects ofeach type (i) belonging to a predetermined set (I) of proj-ects in order to minimize the present worth of the total costof project construction equation (1) subject to constraintson the other two objectives (including achievement of atleast a target amount of land built by 50 years after projectconstruction (equation (2)) and an upper limit on the totalamount of water diverted by all projects (equation (3))),and a requirement that the number of projects be a nonneg-ative integer (equation (4)). In mathematical terms, this is apure linear integer program:

MIN-COST¼ MINni;i2If g

Xi2I

Cini (1)

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subject to:

Xi2I

Lini � LAND (2)

Xi2I

Wini �WATER (3)

ni 2 0; 1; 2; . . .f g; 8i: (4)

[16] The coefficients (in upper case) are defined asfollows:

Ci present worth of construction and operationscost of a project of type i ($);

Li land built by a single project of type i after 50years of operation (km2);

LAND target amount of land to be built (km2);Wi water diverted by a single project of type i

during the high-flow period (m3/s);WATER limit on the water that all projects can divert

during high-flow periods (m3/s).

[17] Several highly simplified assumptions are immedi-ately apparent in this linear model. First, the cost, land, andwater use of each project are independent and are identicalfor projects of the same type. For instance, there are noeconomies of scale in building multiple projects of a giventype, although one might expect efficiencies in design andconstruction. On the other hand, one might also anticipatethat the best sites in terms of cost and land-building poten-tial would be chosen first, so that the incremental cost perproject for a given type might increase with the numberbuilt, while the incremental land built might decrease. Theabove linear formulation assumes that this is not the case.Next, we do not consider critical considerations that coulddifferentiate among candidate sites, such as the effect ofthe location of new land on terrestrial and aquatic ecosys-tems or the potential surge protection benefits. Finally, anyportfolio that uses multiple large diversions needs toaccount for multiple sediment and water mass balances andtheir effects on projects downstream. While the presentanalysis does not specifically account for these interactions,its framework can readily be adapted for that purpose.

[18] Although these simplifying assumptions mean thatthis model cannot be used for actual project planning, ourpurpose is quite different: to understand the general natureof trade-offs between costs and land built, accounting forgeneral-scale economy relationships in project constructionand land building. Such insights can be useful for subsequentsite-specific planning, which can then focus on the types ofalternatives that are robustly in the optimal portfolios andfine-tune them to account for local constraints and concerns.

2.2. Water and Sediment Diversion Model

[19] To quantify the amount of water and sedimentdiverted by a diversion project, which is a necessary inputfor the land-building model, and to provide the basic designspecifications, which is a necessary input for the costmodel, we develop a water and sediment diversion modelbased on a gated box-culvert diversion (Figure 1). Culvertdiversions are one of the three types of engineered diver-sions found on the Mississippi River in Louisiana.The other two types involve tainter gates (e.g., Old River

Auxiliary Control Structure) or low sills (e.g., Old RiverStructure and Bonnet Carr�e Structure). Both tainter gatesand low sill structures divert flow from the main channelby harnessing a significant elevation difference betweenthe water surface in the river and the water surface in thediversion channel. Such conditions occur upstream of NewOrleans where excess water elevations during river floodscan approach 10 m above the adjacent wetland [Nittroueret al., 2012, 2011a]. However, such high water elevationsdo not occur downstream of New Orleans, where riverstage during flood does not exceed approximately 2 mabove the adjacent wetland surface [Nittrouer et al.,2011a]. Under these conditions of relatively low hydraulichead, the gated box-culvert design [Norman et al., 2001]optimizes the potential for diversion of water plus sedimentfrom the main channel.

[20] Sets of large box culverts with submerged outletshave proven to be effective ways of diverting water underat very low head conditions. The diversions at Caernarvonand Davis Pond [U.S. Army Corps of Engineers, 1993;Carter and Bernard, 2004; CPRA, 2007] are cases in point.Large gated culverts are also planned for the proposedMyrtle Grove diversion [Louisiana Coastal Wetlands Con-servation and Restoration Task Force, 2012].

[21] Three parameters characterize an individual box-culvert diversion project (Figure 1): width W, height of thebottom of the diversion opening above the riverbed Zb, andheight of the top of the opening measured from the riverbedZt. The depth D is measured from the bottom of the open-ing to the water surface above the structure. We havedrawn the diversion as consisting of one or more rows ofgated culverts, each with a square 8 m � 8 m opening. Thissize was chosen because it represents an upper limit onstandard unit culverts presently in use or proposed for thelower Mississippi River. The exact dimensions of the unitculvert are, however, not particularly important because ev-ery structure considered here is composed of multiple cul-verts, and it is the sum of their areas that contributes to thecalculated diversion discharge. For example, nearly thesame results as reported here can be achieved using arraysof the 14 ft � 14 ft (4.3 m � 4.3 m) culverts presently inuse at the Davis Pond diversion. The candidate projects theoptimization considers range in size from a single row of 8m culverts to up to three rows of such culverts. Variousnumbers of culverts in each row are considered, as are vari-ous vertical placements, as defined by the parameters Zb

and Zt. Table 1 summarizes the alternatives that we consid-ered, which range from small shallow diversions to wide

Figure 1. Schematic of diversion design parameters.

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very deep diversions. Only individual diversions that wouldtake no more than 45% of the Mississippi high flows wereconsidered (18,000 m3/s), consistent with the Corps limita-tion quoted earlier.

[23] We postulate that these culverts must extend deepinto the flow, rather than skimming off the top, in order toensure optimal capture of sand. Sand is crucial for initiatingland growth especially in the lower Mississippi becausesand (as opposed to mud) settles proximally to the fluvialsource and provides a stable platform for vegetation growth[e.g., Majersky et al., 1997; Kim et al., 2009b]. This vege-tation then facilitates the capture and deposition of mud,which would otherwise be dispersed so broadly that itwould be ineffective for land building [Neph, 2004; Paolaet al., 2011; Nittrouer et al., 2012].

[24] The lower Mississippi River transports both sand(>62 mm in grain diameter) and mud (<62 mm). The sandcan both move along the bed (bedload) and be suspendedin the water column (suspended load). Mud is only trans-ported as suspended load. Previous analyses of the distribu-tion of suspended sediment have considered longitudinaltransport over time [Mossa, 1996]. The model presentedhere, however, emphasizes our more recent understandingof the vertical profile dynamics and the importance of sandcapture for land-building processes.

[25] We assume that sand can be diverted in proportionto concentration in the relevant part of the water column.This may or may not be true. The Mississippi River floodof 2011, however, inadvertently provided an excellent testof the feasibility of diverting sand. During this flood, theBonnet Carr�e Spillway was open for 42 days. Nittrouer etal. [2012] found that 7.8 megatons (MT) of sand, orroughly 40% of the sand load carried by the river duringthis period, was diverted into the floodway. Even thoughthe intent of the diversion was to reduce stage and thereforethe risk of flooding downstream, and not to build land, thisunintended field experiment clearly illustrates the feasibil-ity of diverting large quantities of sand from the channel.The Bonnet Carr�e spillway is a shallow diversion, whichskimmed off only the top 5 m of flow during 2011. Such ashallow diversion was so effective because it is located ad-jacent to a sandy point bar, just downstream of the apex ofa bend [Nittrouer et al., 2012], and this specific site sweptsand into the spillway. However, downstream of New Orle-ans, (a) the river is much deeper, (b) sand bars are muchless developed, and (c) the sinuosity of the river is muchsmaller [Nittrouer et al., 2011b]. In addition, annual floodstages are likely to be significantly lower. These factors

contribute to our conclusion that deep diversions will berequired in order to efficiently divert sand for land-buildingdownstream of New Orleans. Nonetheless, the 2011 experi-ence suggests that a diversion might be designed to besomewhat shallower than the deepest diversions proposedhere, if optimally positioned, for example, adjacent to apersistent alluvial sand bar, where suspended sand concen-trations tend to be enriched. Such optimization could enlistthe help of 3-D numerical models of hydrodynamics andthe spatiotemporal distribution of suspended sediment[Allison and Meselhe, 2010].

[26] In the lower Mississippi River, suspended sand con-centration shows a strong tendency to decline with increas-ing distance z above the bed. For instance, Figure 2reproduces data from Nittrouer et al. [2011a], gatheredfrom three different locations in the Mississippi channelbelow New Orleans (sites S3, S6, and S7 near Empire, Lou-isiana) during the high-flow period in April 2008. Theselocations are in the channel thalweg, where the depth fromthe channel surface to the bottom Dtot is approximately 40m. Figure 2 shows suspended sand concentration in mg/Las a function of percent relative depth below the water sur-face, 100 � (Dtot� z)/Dtot, such that the value 0 corre-sponds to the water surface, and 100 corresponds to theriver bottom. The data show a strong relationship withdepth, with much higher concentrations of suspended loadnear the bottom of the channel. These data are consistentwith both standard and more advanced Rousean models ofequilibrium suspended sediment concentrations in rivers, inwhich upward turbulent transport at any given depth is bal-anced by settling [e.g., Rouse, 1937; Yeh and Parker,2012].

[27] Here we use the form of the standard equilibriumRouse model for the vertical distribution of suspended sedi-ment [Rouse, 1937; U.S. Army Corps of Engineers, 1995]:

Cs zð Þ ¼ Cs;refDtot � z

z

� �Zref

Dtot � Zref

� �� �’; (5)

where z height above bed of river (m);

Cs(z) concentration of sand as function of z (mg/L);Cs,ref concentration of sand at reference height above

riverbed (mg/L);Dtot total depth of river (m);Zref reference height above bed (m);’ Rouse exponent.

Table 1. Dimensions of Candidate Engineered Diversions Considered in Optimizationa

Bottom of Diversion Zb (m Above Riverbed)and Culvert Size (m; in Parentheses)

Width of Diversion W (m)

40 80 120 160 200 240 320 400 480

36 (4 m culverts) 1 1 1 1 1 1 1 1 132 (8 m culverts) 1 1 1 1 1 1 1 1 124 (8 m culverts) 1, 2 1, 2 1, 2 1, 2 1, 2 1, 2 1 1 116 (8 m culverts) 1, 2, 3 1, 2, 3 1, 2, 3 1, 2, 3 1, 2 1, 2 1 1 112 (8 m culverts) 1, 2, 3 1, 2, 3 1, 2, 3 1, 2, 3 1, 2 1, 2 1 1 18 (8 m culverts) 1, 2, 3 1, 2, 3 1, 2, 3 1, 2, 3 1, 2 1, 2 1 1 14 (8 m culverts) 1, 2, 3 1, 2, 3 1, 2, 3 1, 2, 3 1, 2 1, 2 1 1 1

aThe numbers 1–3 in the matrix denote the number of stacked rows of culverts in a scheme. The width of the diversion is specified in the top row, andthe bottom of the diversion and culvert dimension are shown in the left column. Each culvert is assumed to have a square cross section.

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[28] The value of the Rouse exponent equals the ratio ofsettling velocity to the shear velocity, divided by theKarman constant [U.S. Army Corps of Engineers, 1995].Gathering and simplifying the constants in the Rouse equa-tion results in a three-parameter model:

Cs zð Þ ¼ ADtot � z

z

� �’; (6)

where

A ¼ Cs;refZref

Dtot � Zref

� �’: (7)

[29] As the data show in Figure 2, the concentration pro-files are very different at the three sampling sites [Nittroueret al., 2011a]. Of course, more extensive sampling from sev-eral years in locations where structures would be cited wouldbe needed to support actual planning studies, but these andother data in Nittrouer et al. [2011a] show a consistentlystrong relationship of concentration to depth that as we shallsee have important implications for the design of structures.Here we fit these profiles to the form of equation (6) byassuming that the Rouse exponent ’ (which controls the pro-file’s convexity) is identical at the three sites (sites 3, 6, and7), but the constant of proportionality Aj is site dependent:

Cs zð Þ ¼ AjDtot � z

z

� �’; j ¼ S3; S6; S7: (8)

[30] The parameters are estimated as follows. Dtot is 40m, based on the estimated depth of the three sampling loca-tions, and Aj and ’ are estimated from the data shown inFigure 2 using a nonlinear least squares regression. Weassume that the samples are taken from the midpoint ofeach depth interval (0%–20%, 20%–40%, . . . , 80%–100%) reported by Nittrouer et al. [2011a]. The fittedcurves are shown in Figure 2. The resulting estimates are

AS3¼17:5 mg=l;AS6¼91:6 mg=l;AS7¼38:4 mg=l;�¼0:745:

[31] The coefficient of determination R2 of the regres-sion equation is 0.96. Because neither the locations northe depths of the three sampling locations were chosenrandomly, coefficient standard errors cannot be inter-preted in the usual manner. However, as a qualitativeindicator of the precision of the estimate of the Rouseexponent ’, we calculated its standard error using a boot-strap procedure. By this procedure, a large number ofsamples of 15 observations were sampled with replace-ment from those shown in Figure 2, and the Rouse equa-tion was estimated for each of those samples. The resultwas a standard error of 0.2 for ’.

[32] For the purposes of our optimization, we assumethat the concentration profile at the location of a diversionis the average of these three curves, yielding the followingprofile to be used in our sand diversion calculations:

Cs zð Þ ¼ 49:1240� z

z

� �0:745

: (9)

[33] We assume that a given diversion design with anopening that extends vertically from Zb to Zt above theriverbed will draw water uniformly over those depths ;that is, water diversion velocity is the same at all depths.The resulting average concentration is obtained bynumerical integration of equation (9) over those depths.As with many of our other assumptions, more sophisti-cated assumptions could be made about the verticalprofile of water diversion velocities, but they would notchange the qualitative nature of our results. The key phe-nomenon to be captured is that concentration increaseswith depth below the water surface, so that deeper diver-sions will provide more sand for a given quantity ofwater diverted.

[34] The water discharge through a given diversion wasspecified using standard culvert design techniques[Norman et al., 2001]. The culvert is assumed to be fullysubmerged at both the headwater and tailwater locations.The discharge Qc through the set of culverts shown inFigure 1 is

Qc ¼ W Zt � Zbð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g �r � �tð Þ

1þ �e þ �e�2�=�4=3

s; (10)

where �r water surface elevation in the river adjacent tothe diversion (m);�t tailwater surface elevation just downstream of the

diversion (m);g gravitational acceleration (m2/s);� culvert length (m);� hydraulic radius of each culvert (m);� Manning’s ‘‘n’’ coefficient of resistance;�u standard culvert coefficient ;�e entrance loss coefficient.

[35] The following representative values were used inthe calculations below; �r¼ 2 m, �t¼ 0 m (mean sea level),�¼ 100 m, �¼ 0.02, �u¼ 19.63 (standard value for metricMKS units), �e¼ 0.5, and �¼ 1.52 m. These values can beeasily modified for any site-specific calculation. The

Figure 2. Suspended sand concentration sampled fromEmpire Reach of Mississippi River in deepest part of thechannel (sites S3, S6, S7) [Nittrouer et al., 2011a, Appen-dix A] and fitted Rouse profiles.

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discharge of sand through the diversion Qsand (in t/s) wasthen computed as

Qsand ¼ 1� 10�6QcCs: (11)

2.3. Land-Building Model

[36] One of the three objectives included in the optimiza-tion model is the amount of land that can be built, given aparticular diversion project design, after 50 years. Thisstudy uses a modified version of a well-established land-building model by Kim et al. [2009a] in which they adaptedpreviously developed numerical models of delta evolution[e.g., Parker et al., 1998; Kostic and Parker, 2003; Kim etal., 2009b] specifically for land building by MississippiRiver diversions. In the model, specified discharges ofwater and sand are diverted from the river and allowed tospread out over a specified bathymetry, subject to specifiedrates of subsidence and sea level rise. Morphodynamicinteractions among the flow, sand transport, and morphol-ogy results in the formation of a fan-shaped delta, of whichdeltaic surface aggrades (builds upward) and shoreline pro-grades (builds seaward over the preexisting bathymetry).The new land built this way provides a platform for thegrowth of vegetation, increased efficiency of capture ofmud from the flow, the formation of organic soil, and thedevelopment of new wetlands habitat [Neph, 2004; Paolaet al., 2011; Nittrouer et al., 2012; Day et al., 2012].

[37] Kim et al. [2009b] tested and calibrated their land-building model based on the observed evolution of the WaxLake Delta. This delta resulted from the capture of part ofthe flow in the Atchafalaya River by an engineered channelknown as the Wax Lake Outlet. The Atchafalaya River isin turn a distributary of the Mississippi River itself. A deltabegan to build seaward from the mouth of the Wax LakeOutlet in 1980. It has been extensively monitored subse-quently [e.g., Roberts et al., 2003] and as such serves as anatural laboratory for the testing of land-building models.The land-building model used a range of sediment dis-charges to the outlet between the maximum (38 MT/yr)and minimum (25 MT/yr). The modeling results from thesetwo end members bracket the range of observed deltagrowth history [Kim et al., 2009b; Paola et al., 2011].Upon testing against the Wax Lake Delta, Kim et al.[2009b] applied the model to a pair of hypothetical diver-sions below New Orleans, one into Breton Sound and theother into Barataria Bay, and showed how diverting 45% offlood flow could build around 900 km2 of new land in acentury. Kim et al. [2009b] also applied appropriate rangesof sea level rise rates and subsidence rates to test sensitivityof the model and different scenarios on land building to thediversion sites. The sea level rise rates were 0–10 mm/yr,and the subsidence rates were 1–4 mm/yr, providing a 700–1200 km2 of new building land over the next century [Kimet al., 2009a, 2009b].

[38] The model is capable of accounting for various pa-rameters, details of which can be found in Kim et al.[2009b]. These include the rates of sea level rise and subsi-dence, as we discussed earlier, as well as the characteristicgrain size of the sand, antecedent bathymetry, efficiency ofcapture of mud, and fraction of time the river is in flood,which can be easily adapted to be site specific. However,

the main point in this paper is to examine the relationshipbetween size and cost of the diversions and land building.Therefore, the parameters to be considered here, except forthe discharges of water and sand, are set equal to the valuesfor Barataria Bay and applied over the next 50 years (toKim et al. [2009b, supplementary material] ; www.agu.org/pubs/eos-news/supplements/2009/kim_90_42.shtml).

[39] The land-building model, which has been testedagainst the deltaic systems in the Mississippi River Basinunder a wide range of environmental conditions, is appliedto predict the area of new land created over a given period,given water, and sand discharges. These discharges them-selves are obtained from the water and sediment diversionmodel described earlier. The results of the land-buildingmodel are linked to the diversion cost model below.

[40] A statistical analysis of the relationship of theamount of land built by 50 years to the annual quantity ofwater and sand diverted under the parameters assumed ear-lier shows that an exponential function provides an excel-lent fit to the results. In particular,

Land km 2� �

¼ 63:54 Sand t=yrð Þð Þ0:7715; R2 ¼ 0:999: (12)

[41] Since the exponent is less than one, the functionexhibits moderate-scale diseconomies for land built as afunction of sand, as Figure 3 shows. The reason is the ba-thymetry, i.e., increasing depth with distance from thediversion, so more sediment is needed for incremental landadditions for larger deltas. Although there is no water termin the above regression, the simulation model does showthat if more water is added, slightly more land is built in allcases; however, less than 0.001% more land results if 1%more water is added. The water term is not statistically sig-nificant and so is omitted from equation (12).

[42] The regression shows that doubling the amount ofsand increases the amount of land built by year 50 by only73%. Note, however, that the optimization model equations(1)–(4) use the original land-building estimates for individ-ual projects, and not estimates resulting from the approxi-mation equation (12). This is important because (12) is notexact for all project scales. In particular, for smaller proj-ects, the relationship is more strongly concave, with signifi-cantly less than 73% more land built when sand is doubled,

Figure 3. Fitted function for land as a function of sanddiverted showing scale diseconomies.

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which, as we will explain, has important implications forthe optimal width of shallow diversions.

2.4. Cost Model

[43] To estimate the cost of constructing a diversiongiven the depth and width of the diversion, we developed acost model. This general model is parsimonious so that itsfew parameters can be thoroughly explored in respect tohow alternative assumptions about the degree of scaleeconomies or diseconomies in construction affect our con-clusions. The function is

Cost ¼�ðD=HÞW ; (13)

where depth D is normalized by the vertical distance Hfrom the highest water level to the river bottom, and W isthe width of the diversion, as defined earlier in this paper.

[44] The cost function is an exponential or ‘‘Cobb-Doug-las’’ function, which is commonly applied in mechanicaland chemical engineering to express the relationshipbetween structural capacity and cost [Cobb and Douglas,1928]. This compact specification has the advantage thatcosts are zero when the diversion opening has a zero cross-sectional area (i.e., when either or both W and D are zero).It also captures the nonlinear nature of costs, and the expo-nents are readily interpreted as representing the degree ofscale economies or diseconomies.

[45] In this analysis, all proposed system costs areexpressed as a percentage of the most expensive systemconsidered, rather than in absolute dollar values in order toemphasize how the exponents and , which control scaleeconomies/diseconomies, affect the choice of systemdesign. Thus, the scaling constant � is unimportant, and wedisregard it in the analyses.

[46] It is expected that project construction will showscale economies in width W, such that doubling widthwould result in higher costs but less than a full doubling ofcost. Although land requirements could reasonably beexpected to increase in proportion to width, scaleeconomies might characterize the expense of civil works,including foundations and structures. Thus, we hypothesizethat 0<� 1. On the other hand, there are likely to bescale diseconomies with respect to depth D, because deeperopenings would, for example, likely require considerablymore substantial abutments and complex gate control struc-tures. Thus, we conjecture that � 1. We consider severalcombinations of parameters that satisfy those restrictions,with scale effects ranging from negligible to large.

[47] Evidence for these hypotheses is construction costdata for existing Mississippi River delta diversion

structures. Table 2 presents the cost and dimensions of fiveexisting diversions; their sizes encompass the range ofdiversions being considered in the Louisiana CoastalProtection and Restoration Authority of Louisiana study[State of Louisiana, 2012a], with the largest diversions con-sidered being about the same size in terms of water quanti-ties (approximately 7000 m3/s) as the existing BonnetCarr�e diversion. The construction costs in Table 2 wereescalated to end of year 2009$ using the levee and flood-wall cost indices from 1980 to 2009 from U.S. Army Corpsof Engineers [2011] and the Engineering News Record[2010] construction cost indices for previous years.

[49] Fitting Cost to the 2009$ costs in Table 2 results inthe following estimated cost function:

Cost ¼ 0:427 D=Hð Þ1:64W 0:48; R2 ¼ 0:997: (14)

[50] This estimation only considers construction costs andnot the additional costs from operations and maintenance. Ifwe make the simple but not implausible assumption that suchcosts are roughly proportional to construction cost (which isalso the assumption that the master plan makes about diver-sions [State of Louisiana, 2012b], which assumed that 50years of operations and maintenance (O&M) amount to 18%of diversion investment costs), then including them wouldnot affect the values of the exponents, which are our focus.The magnitude of the costs for the larger structures in the ta-ble is broadly consistent with those in the master plan [Stateof Louisiana, 2012b] (see discussion in section 3.1).

[51] The excellent model fit is at least in part becausefew data points were used in the estimation. Nonetheless,the results provide evidence for our hypothesis of scaleeconomies in width and scale diseconomies in depth. Thecosts of large diversions are uncertain because none havebeen built recently and their design may differ appreciablyfrom past diversion structures.

[52] The scale economies for width estimated by theabove procedure are stronger than assumed in the masterplan [State of Louisiana, 2012b]. The costs of alternativediversions in the master plan were developed by scaling theanticipated channel construction costs of the proposedMyrtle Grove diversion using the dimensions of the chan-nels required at different locations and adjusting for localcosts of moving highways (N. Snider, personal communi-cation). The cost of the diversion structure in each case wasassumed to be proportional to the width, which in turn wasassumed to be proportional to the maximum flow diverted.Thus, the master plan assumed zero-scale economies inwidth, in contrast to the results of our statistical analysis. Afixed depth was assumed for all diversions, unlike our

Table 2. Cost, Depth, and Width of Diversion Projects in the Mississippi River Delta

Diversion Project

Original StructureCost (2009$in millions) Completion Date

Cost (2009$in millions)

DepthRatio D/H

DiversionWidth W (m)

Predicted Cost(2009$ in millions),

equation (14)

Bonnet Carr�e 13.3 1929 480.9 0.22 2330 480.5Caernarvon Diversion 26.0 1991 46.3 0.17 57 76.2Davis Pond Freshwater Diversion 129 2010 129.0 0.22 74 98.4Old River Control Structure 273 1963 ($67M),

1986 ($206M)988.5 0.8 425 989.1

West Bay Sediment Diversion 4.54 2004 5.92 0.15 170 21.4

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analysis that considers alternative depths as well as alterna-tive assumptions concerning the anticipated costs of diver-sion structures.

[53] These results motivate the consideration of the fol-lowing ranges of the scale parameters : � [1,1.6] (repre-senting a range from no-scale effects for depth to strong-scale diseconomies) and � [0.6,1] (ranging from strong-scale economies for width to no-scale effects). As will beshown, the strength of the cost-scale economies/disecono-mies has a significant effect on project choice, at least forthe case where relatively low amounts of land are built.

[54] The above cost function has considered only depthand width of the diversion project and not the height of thegated opening itself (Zt�Zb, Figure 1). This assumption isbecause we have no data for how that height affects thecost of a structure like Figure 1 for a given W and D. There-fore, we consider three possibilities here: that the numberof rows of 8 m � 8 m culverts (which determines the heightof the opening) has no effect on cost (base case), and twosensitivity cases in which each additional row of culvertsabove 1 row adds either 10% or 20% to the project cost.The larger value would represent approximately a doublingof the cost of the gate structures, which according to Stateof Louisiana estimates, account for 20% or less of the costsof the larger gated diversions considered in the master plan(N. Snider, personal communication). These costs could berefined for a site-specific study by detailed engineeringstudies of alternative structure designs, but this analysis isbeyond the scope of this paper.

3. Results

[55] The above sequence of models of sand and waterdiversions, land building, and cost allows us to analyzescale economies and diseconomies for individual projectsas well as for portfolios of projects. Mathematically, a port-folio that is efficient in terms of the three major objectives(minimize cost, maximize land built in 50 years, and mini-mize water diverted) must consist of projects that are indi-vidually efficient for those objectives (i.e., there exists noother project that does better in any objective while beingno worse in the others). This can be shown by contradic-tion: if an efficient portfolio included an inefficient project,then a project that dominates the inefficient project (i.e., isbetter in at least one objective and no worse in the others)could be substituted, thereby resulting in a new portfoliothat dominates the original portfolio, which is acontradiction.

[56] Therefore, we first explore how the set of individu-ally efficient projects from among the candidate designs inTable 1 depends upon assumptions concerning the degreeof scale effects in equation (13). Although the results ourcost analysis indicate that wider diversions are less expen-sive per unit of width and deeper diversions are more ex-pensive to build per unit depth, we consider two polarscenarios concerning scale economies : one with no-scaleeconomies in diversion width or depth and the other withstrong-scale effects. Second, we use the optimization modelof section 2.1 to assemble efficient individual diversionprojects into efficient portfolios in section 3.2 under thetwo polar-scale economy assumptions. Details on the indi-

vidual efficient projects and portfolios results are providedin Appendices A and B, respectively.

3.1. Single Diversion Land-Building Results

[57] The performance of individual projects in terms ofthe cost and land objectives is presented in Figure 4. Pro-ject costs (equation (13)) are scaled relative to the largestproject considered, which has a cost of 100%. The symbolsindicate whether the projects are shallow, medium, or deep(depth to bottom of opening of <10, 10–25, and> 25m,respectively) and the fraction of flow diverted (0%–12.5%,12.5%–25%, and 25%–45%, respectively). Figure 4a showsthe no-scale economies case (indicating that cost increasesin proportion to width and depth), setting ¼ ¼ 1, whilethe strong-scale effects case in Figure 4b has ¼ 1.6 (dou-bling depth D more than doubles cost) and ¼ 0.6 (dou-bling width W less than doubles cost).

[58] The most immediately apparent results in Figures 4aand 4b are that an individual project can build largeamounts of land only if it is deep; the points to the right ofFigure 4 (indicating more land) are the deepest projects.

Figure 4. Land-cost characteristics of individual projectswhen cost functions exhibit : (a) no-scale economies or dis-economies (¼ ¼ 1 in equation (13)) and (b) strong dise-conomies in depth D and strong-scale economies in widthW (¼ 1.6, ¼ 0.6 in equation (13)). The shape symbolsindicate whether the projects are shallow, medium, or deep(depth to bottom of opening of <10, 10–25, and >25 m,respectively) and the fraction of flow diverted, as indicatedby the gray-scale coloration from white to gray to black(0%–12.5%, 12.5%–25%, and 25%–45%, respectively).

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This result occurs because of the higher concentrations ofsand at depth and the limitation upon the maximum amountof water diverted (45% of flow). Specifically, deeper diver-sions deliver more sand per unit water (Figure 2), which forinstance, shows that concentrations of sand for depths thatare 20%–40% below the surface carry about a quarter ofthe sand found at depths 60%–80% below the river surface.In theory, it is possible to build a wider shallow project todivert enough sand to build as much land as a deeper butnarrower project, but the water diversion limitation(designed to protect navigation and water supply) rules outvery wide and shallow projects. Thus, to increase theamount of land built for a given amount of water, it is nec-essary to increase the sediment-to-water ratio, which canbe accomplished by pulling flow from greater depths in thewater column.

[59] To put those cost numbers in perspective, the proj-ects in Figure 4 can be compared to the historical projectswe considered in Table 2 as well as the anticipated cost andland for the five largest candidate diversion projects in theLouisiana Coastal Master Plan [State of Louisiana, 2012a].First, considering the historical projects in Table 2, they lieprimarily in the extremes: a very large, deep project (OldRiver Control Structure), a very wide shallow project(Bonnet Carr�e), and three small, shallow projects. If exist-ing projects were to be mapped on Figure 4, the Old RiverControl Structure (which was developed for flood controlnot land building) would appear on the upper right quadrantof this figure. This would mean that approximately $1billion (the 2009$ cost of that structure) would correspondto 100% on the cost axis.

[60] Meanwhile, a somewhat higher cost estimate resultsif we instead compare these projects with those consideredin the Louisiana Master Plan. Each of the five largestdiversions in the plan would divert 250,000 cfs (7080 m3/s)during the flood season for land building, are anticipated tohave a capital cost in the range of $0.6–1.1 billion (aboutthe same as the Old River Control Structure, Table 2), andwould build approximately 75–280 km2 by year 42 afterproject completion [State of Louisiana, 2012a]. (The masterplan reports land values for 50 years after project planningbegins or 42 years after the start of project operation.) As arough approximation, this can be used to assign estimatedconstruction cost values to the cost results shown in Figure4. Using the midvalues of those ranges and assuming that15% more land would be built after 50 years of operationresult in estimates of $0.85 billion for costs and 200 km2

for land. Additionally, assuming that these projects areeither medium or shallow in depth (less than 25 m deep)would mean that such a project would cost between 30%and 50% of the cost of the largest projects considered inFigure 4. This would imply that the most expensive proj-ects in Figure 4 would cost between $1.7 billion and $2.8billion, which is somewhat larger than the $1 billion valueimplied by the Old River Control Structure.

[61] From a 2-D figure such as Figure 4, it is impossibleto identify all the efficient single projects in terms of allthree objectives (cost, land, and water). Those on the lowerenvelope are all efficient. However, some points withhigher costs (above the envelope) are also efficient becausethey use less water while building the same amount of landas points below them. Appendix A lists all the efficient

individual project designs. In general, there are a greaternumber of efficient options when there are strong-scaleeconomies (Figure 4b and Appendix A) than when thereare none (Figure 4a and Appendix A). All designs that areefficient for the latter case are also efficient in the former.The reason why there are a greater number of efficient proj-ects under strong-scale economies is that some smallershallow projects become more cost-advantageous relativeto deep projects, making up for their inefficiently low ratioof sand to water.

[62] There are also patterns in the type of projects thattend to be the least costly means of building a targetamount of land (the lower envelope of points in Figure 4).Assuming no-scale economies (Figure 4a), deeper projectsare always more cost-effective than shallow ones (nearlyall the lower points are circles, indicating deep projects,while none are the triangles that designate shallow proj-ects). This result should not be surprising because no-scaleeconomies imply that for a given size of opening, the costwill be the same for a deep and narrow project as it wouldbe for a wide and shallow project, but the deeper projectwill access more sand.

[63] However, if instead strong-scale effects areassumed, consistent with our analysis of already con-structed diversions (Figure 4b), a 1% increase in widthincreases cost by 0.6%, but a 1% increase in depthincreases cost by 1.6%. Nevertheless, except for the small-est amounts of land (<80 km2), shallow projects are neverthe cheapest way to build land because deep projects havea higher sand-to-water ratio (Figure 2), which compensatesfor the more expensive cost of the diversion opening. Deepprojects are the most cost-effective way to build land whenapproximately 200 km2 or more land is being built with asingle project ; projects of medium depth dominate fromabout 100 to 200 km2.

[64] Additionally, it is important to note that the some-times convex shape of the lower envelope, in which thewidest, shallow project builds more land per $ than otherprojects, means that in the absence of the water constraint,several smaller, shallow projects would be a cheaper wayto build land than a single larger and deeper project underthose cost assumptions (as in Turner and Boyer [1997]).However, such a strategy is infeasible because shallowerprojects use too much water when building large amountsof land, and attempting to reach a 200 km2 target in thismanner would require more than 20,000 m3/s of water tobe diverted in total.

3.2. Portfolios of Diversions

[65] In general, our results show that the most efficientway to use the water and sand available to build largeamounts of land can be to build one or a few large projectsrather than build several small or medium size projects.The extent to which this result is true depends on (1) thetrade-offs between scale economies in project construction(wider diversions are cheaper per unit (equation (14) andFigure 2), (2) the relative rates at which costs and sandconcentrations increase with depth (equation (14) andFigure 2), and (3) the scale diseconomies in land building(doubling sand diversions yields less than double theamount of land, Figure 3). Here we apply the optimization

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model of section 2.1 to assemble portfolios of projects thatare efficient in terms of the land, cost, and water objectives.

[66] Figure 5 shows alternative efficient portfolios plot-ted so that the trade-offs among land, cost, and water areapparent. The plots include two different cost functions:no-scale effects (¼ ¼ 1 in equation (13); Figure 5a) andlarge-scale economies in project width ¼ 0.6 togetherwith large-scale diseconomies in project depth ¼ 1.6(Figure 5b). Figures 5a and 5b can be interpreted as fol-lows. Each point is an efficient portfolio of one or moreprojects such that no other portfolio simultaneously costsless and uses less water, while producing more land. Differ-ent points represent trade-offs (for instance, going from onepoint to another that uses more water for a given amount ofland will lower cost). Not all efficient portfolios are shown;rather, these are a sample that illustrates the trade-offsamong the three objectives.

[67] The efficient sets are nonlinear ; regression analysisshows that cost is proportional to LAND1.5WATER�0.5

(R2¼ 0.99) if there are no-scale economies in cost, and toLAND1.4WATER�0.5 (R2¼ 0.98) if there are scale effects

in cost. Thus, the overall cost of land is convex (increasingcost per unit as more land is built, given a fixed amount ofwater use), while making more water available for diver-sions will lower costs, but at a diminishing rate. In particu-lar, for a given amount of water diverted, increasing theland target by 10% increases the cost by an average of14%–15% (depending on whether scale economies areassumed or not). Meanwhile, increasing the amount ofwater available to meet a given land target by 10%decreases cost by an average of 5%.

[68] The lines on the water-land plane show the projec-tion of the edges of the efficient set to that plane. The upperright edge is defined by portfolios that maximize the landbuilt for a given amount of water diverted, irrespective ofcost. That line is the same for both cost functions becausecost is irrelevant to that maximization; no collection ofprojects can create more land with the given amount ofwater. This edge shows that for example, increasing theallowed water diversion from, say, 18,000 to 25,000 m3/sresults in a proportional increase in the maximum land thatcan be built in 50 years (from 1000 to 1400 km2). In

Figure 5. Efficient set of portfolios in objective space (land (km2)-water (m2/s)-cost (% of highest costportfolio)) for (a) noscale economies and diseconomies of project costs and (b) strong-scale economiesin width and diseconomies in depth. Note that (a) and (b) do not include a representative set of the effi-cient portfolios, and because of the discrete nature of the projects, the noninferior set is necessarily a col-lection of points.

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contrast, the lower edge is affected by the cost-scale econ-omy assumptions, being defined by portfolios that are theleast-cost means of achieving a land target without regardto the water consumed. In the case of strong-scale econo-mies, Figure 5b shows that the least-cost means of building200 km2 uses 25,000 m3/s, while the cost function with no-scale effects implies that the least-cost means of achievingthat land target would require only 8000 m3/s (Figure 5a).The third (left) edge is defined by efficient portfolios thatminimize the cost of achieving the land target, but bumpagainst a total water use constraint of 25,000 m3/s. Portfo-lios requiring more water are not considered because it is40% more than the master plan allows [State of Louisiana,2012a]. Finally, points among the three lines represent in-termediate values of land, cost, and water use.

[69] Figures 6a and 6b (all efficient portfolios aredetailed in Appendix B) provide details on the projects thatmake up efficient portfolios for a given land-building goal.These portfolios are the least-cost solutions of model (1)–(4) subject to land targets between 100 and 700 km2 and alimit of 18,000 m3/s for water use (100% of water limit).Thus, these are points that lie along the lower line in Figure5, following that line as the land target increases until the

water limit is met ; the subsequent solutions then parallelthe land axis so as to maintain water use at that level. Fig-ure 6 shows that with one exception noted later, these effi-cient portfolios include at least one deep project, with deepprojects building the bulk of land in each portfolio. Thereare two reasons for this result ; one is that the increase insand with more depth makes up for the increased cost ofdeeper projects, and the second is that the water constraintbecomes binding for larger land targets, requiring use ofdeep sand-laden water.

[70] The exception to this conclusion is the portfolio thatbuilds the smallest amount of land (100 km2) in the strong-scale effects case (Figure 6b and Appendix B), which con-sists of many shallow, narrow diversions. Cost savingsfrom building shallow diversions more than compensate forthe relatively low sand concentrations in the water that theydivert, and the upper bound on diverted water does not yetbind. However, such a replication strategy is infeasiblebecause the water constraint quickly becomes binding.Even for building just 100 km2 of land, such a strategydiverts over two thirds of the allowed 18,000 m3/s of water.This means that for 200 km2 or more land, cheap shallowprojects divert too much water, so that water must be con-served by building costly deep projects. In contrast, for thecase of no-scale economies, Figure 4a shows that deepprojects are much cheaper per unit of land, so indeed theymake up all the optimal portfolios under this cost function.Thus, under all cost functions, if the goal is to cost effec-tively build moderate to large amounts of land, it is essen-tial to build deep projects because the water constraintbecomes binding, forcing diversion designs to take deeper,sand-rich water.

[71] The assumption about scale economies with respectto width highlights an important difference in the portfolioswith the largest amount of the land. For no-scale economies(Figures 5a and 6a and Appendix B), to build 700 km2 ofland, the optimal decision is to build several deep but rela-tively narrow projects. This result occurs because of the ba-thymetry effect in which the edge of larger deltas lies indeeper water, thus requiring more sand per unit of incre-mental land built (i.e., diseconomies in building land; Fig-ure 3). In contrast, in the situation with strong-scaleeconomies in width, fewer and wider projects are optimalfor large land targets (Figures 5b and 6b and Appendix B).The optimality of many narrow projects versus few wideprojects hinges critically on the relative value of the expo-nent for width in the cost equation (14) versus the exponentfor sand in the land-sand function (equation (12) and Figure3). In Figure 6b, cost per unit width falls faster than theland per unit sand (exponents of 0.6 and 0.7715, respec-tively), so a single, wide project will produce more land perunit cost than several narrower projects.

[72] To compare with the above solutions, an extremeportfolio was derived that creates the maximum amount ofland given the water limit, ignoring project costs (upperedge in Figure 5). In 50 years, the maximum amount ofland that could be built under a water limit of 18,000 m3/sis 980 km2, which is approximately 8% of the current deltaland area (assuming a current footprint of about 12,700km2) [Louisiana Coastal Master Plan, 2012, p. 103]. Thisamount of land is built with 12 deep and narrow projects(Zb¼ 4 m, Zt¼ 12 m, W¼ 40 m; Table A1). These projects

Figure 6. Optimal set of diversion projects to achieve atarget area of land built in 50 years, given an 18,000 m3/swater limit and (a) no-scale economies and diseconomiesand (b) strong scale of economies in width and disecono-mies in depth. The project indicated by � provides the bulkof land in the portfolio when portfolios include multipleproject types. The shallow projects have a depth to bottomof opening of <10 m, medium 10–25 m, and deep >25 m.Wide projects exceed 100 m in width, while narrow proj-ects are below that threshold.

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maximize the amount of land built under the water con-straint because they capture a high sand-to-water ratio andare able to disperse that land over several smaller deltas tomaximize the amount of land built with a given amount ofsand (by minimizing the bathymetric effect that causes theconcavity of the land-sand relationship, Figure 3).

3.3. Sensitivity Analyses of Portfolios

[73] The sensitivity of the results with respect to severalassumptions was also tested, including (1) changes inscale-economy assumptions, (2) limiting the size of indi-vidual projects (as measured by water diverted), and (3)imposition of a 10% cost premium for a project with twostacks of 8 m culverts rather than one (Figure 1) and a 20%premium for a three-stack project.

[74] First, we consider how the results change if scaleeffects are assumed other than the two polar cases ofno-scale effects in both width and depth and strong-scaleeffects (¼ 0.6 and ¼ 1.6). We conducted sensitivityanalyses of the effect of other mixes and degrees of scaleeffects upon the optimal portfolios, such as scale effects inwidth only (e.g., ¼ 0.6, ¼ 1) or depth only (e.g., ¼ 1,¼ 1.4), or moderate levels of both (e.g., ¼ 0.8, ¼ 1.4).The following trends in the results reinforce our generalconclusions. First, no matter what degree of scale effectsare assumed, in order to build relative large amounts ofland (700 km2 in 50 years) under a water limitation of18,000 m3/s, the efficient portfolio would be primarily orentirely comprised of projects that are as deep as possible.Second, no matter the assumption of the degree of scaleeffects for depth, if scale economies in width are absent(¼ 1), then it is better to build three to four fairly equallysized projects because smaller projects (for the same depth)build more land per unit of water diverted. On the otherhand, if scale economies in width are present (¼ 0.6 or0.8), then the project portfolio would be dominated by asingle very wide project taking up about half of the budget,complemented by approximately three considerablysmaller projects using up the rest of the budget.

[75] Second, we consider how the optimal project mixwould change if we tightened the water constraint for anindividual project to 4000 m3/s or 8000 m3/s (base case-¼ 18,000 m3/s). This represents a situation in which envi-ronmental impacts or social concern with a larger singleproject makes them politically infeasible. These tighterconstraints serve to exclude some of the largest projects inTable 1 from consideration.

[76] There is no difference between the base case(Figure 6) and sensitivity analysis cases for the 8000 m3/slimit case under no-scale economies and an 18,000 m3/slimit. For the 4000 m3/s case without scale economies, thelarger projects formerly used in the portfolio are not possi-ble. Projects involving three rows of culverts were formerlyused; now projects with two rows are substituted in orderto divert less water. The result is a 10%–20% increase incost for building land in the range of 200–500 km2. For thestrong-scale economies case, for the 8000 m3/s waterconstraint, the land-building portfolios between 100 and600 km2 consist of double the number of projects as in thebase case, with much narrower widths necessary to meetthe water constraint. Costs increase by 10%–20% for the300–500 km2 land-building portfolios. Under a 4000 m3/s

limit on individual projects, the number of projects in theoptimal portfolios for 300 km2 of land or more rises tobetween 6 and 16 projects. When narrower projects arebuilt, scale economies are lost, and the portfolio costsincreases by 20%–25% for building 300–600 km2. Clearly,prohibiting large projects imposes a significant cost penaltyfor achieving a given land target.

[77] Third, the base case assumed that diversions withthe same bottom depth and width have the same costswhether they have one, two, or three rows of culverts. Theeffect of this assumption was explored by adding a costpremium of 10% for two stacks and 20% for three stacks.For both cost functions and for a minority of the portfolios,there was a shift from three-row projects to two-rowprojects, or from two-row to single-row projects. However,the general conclusions do not change.

4. Discussion and Conclusions

4.1. Discussion

[78] This study is the first to rigorously couple land-building predictions from water and sediment diversionswith a cost analysis. Using an optimization framework, thisstudy defines portfolios of diversion projects consideringcumulative water and land effects, variable sediment con-centration in the water column, and scale effects in landbuilding and costs for optimal project mixes. Our resultsprovide the context for management decisions (i.e., thelikely feasible and cost-effective set). Specifically, if thepolicy target is to build 200 km2 of land or more, it is notpossible to meet such a target without at least one deepdiversion project. For portfolios, most of the land is builtwith one deep project if there are strong-scale economies inwidth and diseconomies in depth. For the case of no- tomoderate-scale economies/diseconomies in width/depth,respectively, multiple deep but narrow projects build themost land for the least cost because of diminishing returnsin terms of land built per unit sand. This is due to bathyme-try and the water constraint. Deep projects use water moreefficiently (greater sediment delivery per unit water), and ifthere are no- to small-scale diseconomies with depth, thenthe cost per unit of land built is comparable or even cheaperthan the cost of land from shallow projects.

[79] The maximizing land-building scenario, whichbuilds 980 km2 if not more than 18,000 m3/s of water canbe diverted, is consistent with the State of Louisiana[2012a] coastal restoration master plan that predicts thatthe diversion projects will build approximately 900 km2

over 42 years. Our result is equivalent to an average annualland-building rate of about 20 km2/yr (980 km2/50 yr),which approaches the rate of wetland loss of approximately24 km2/yr reported by Morton et al. [2005a] for the years1990–2000. To build this amount of land requires approxi-mately 17 MT/yr of sand, which is about 80% of the annualsand load and includes an equal amount of mud. Therefore,the total amount of sediment diverted per year to build landis about 30% of the overall annual sediment budget for thelower Mississippi River of approximately 126 MT/yr.

[80] What we assumed for the most water that can bediverted strongly affected the portfolios and amount of landthat can be built, as shown in Figure 5. For instance, if wechanged the water limit from 18,000 to 11,500 m3/s, the

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most land that could be built falls from 980 to 520 km2.Thus, the portfolios in Figure 6 that build 600 and 700 km2

are now infeasible. The other portfolios are generally thesame, shifting in some cases to configurations that usesomewhat less water. If the water limit is raised to 25,000m3/s, it now becomes feasible to build 1300 km2 of land.This case should be looked at as an upper limit, requiringdischarges on the lowermost Mississippi River roughlyconnected with the 10 year flood. In this case the amount ofsand diverted would amount to approximately 1.6 m3/s orbetween 25 and 30 MT/yr, corresponding to most of theaverage annual sand load in the Mississippi River belowthe Old River Structure. The general conclusion that deepdiversions are required to build significant amounts of landis unaffected, as is the conclusion that strong-scale effectsin the cost function favor fewer and wider projects.

[81] Because the conclusions of this research disagreewith those of Turner and Boyer [1997], we compare ourcosts to those of portfolios using their optimal project scale.Using a scenario with small-scale projects that are 4 mdeep and 40 m wide, the water constraint limits the maxi-mum number of projects to 25. These 25 projects wouldonly build 139 km2 of land and would cost 0% (¼ 1.6,¼ 0.6) to 52% (¼ 1, ¼ 1) more than the optimal basecase portfolio that would build the equivalent amount ofland. Not only is this approach not the most cost-efficientmethod of building land, but it is also on the lower end offeasible alternatives.

[82] However, shallow projects can be cost-effective inbuilding small amounts of land, if water limitations are nota constraint and there are strong-scale diseconomies withdepth. In spite of this, the decision to rely upon shallowprojects could preclude adoption of more ambitious futureland targets because of the water constraint, unless thesmaller-scale projects are later abandoned to make way forlarge projects.

4.2. Conclusions

[83] The analysis is general in nature, and thus, it isnot of sufficient detail for site-specific project recommen-dations. However, the framework could be adapted forthat purpose. For example, real potential sites will becharacterized by various antecedent bathymetry profilesand to a smaller degree, different rates of subsidence.This notwithstanding application of the model of Kim etal. [2009b] to three different sites, i.e., the Wax LakeDelta, the diversion to Barataria Bay, and the diversionof Breton Sound, suggests that the generic model pre-sented here is quite sufficient for general estimates of rel-ative cost, regardless of the details of the site in question.Although we use a general model to develop conclusionsacross the Mississippi delta landscape, there are spatiallyexplicit numerical and physical models that consider local

conditions [Willson et al., 2007; Karadogan and Willson,2010]. Any specific candidate for a site should be subjectto such site-specific analyses, adapted in detail to the rel-evant conditions.

[84] Additionally, this study used a generic cost functionequation (13) developed using existing diversions. Depend-ing on the location, the cost could be higher or lower thanin equation (13) because of local foundation and environ-mental or societal factors. A planning study would involvedetailed engineering design work to specify structuredesign and estimate the resulting cost, given the location.Our generic cost function, however, sufficed to show therobustness of our conclusions to the extent of scale econo-mies or diseconomies.

[85] Given the importance of coastal restoration for thedelivery of ecosystem services and storm surge protection,it is essential to carefully assess how land-building goalscan be achieved cost-effectively. Despite the large numberof planning studies in the Mississippi delta in the last twodecades, none prior to the 2012 master plan [State ofLouisiana, 2012a] have addressed what scale of projects ismost efficient. Unlike the master plan, our study considersthe implications of alternative scale and other assumptionsfor the identification efficient portfolios of projects, allow-ing conclusions that can be generalized to the delta. Eventhough our analysis cannot be used for site-specific projectrecommendations, it demonstrates that for the most ambi-tious 50 year land-building goals, it is necessary to capturea higher sediment-to-water ratio by building deep diversionprojects. Consistent with the conclusions of our analysis,the master plan concludes that ‘‘(b)ecause [sediment diver-sions] are so effective, it is no longer a question of whetherwe will do large-scale diversions but how we will dothem.’’

Appendix A: Single Project Results

[86] Tables A1–A4 provide dimensional and perform-ance information for individual projects. Tables A1–A3summarize projects that are single, double, or triple stacksof 8 m culverts, respectively, while Table A4 shows proj-ects that are single stacks of 4 m culverts.

Appendix B: Portfolio of Project Results

[88] Tables B1 and B2 summarize selected portfolios ofprojects that are efficient in terms of the three objectives.Table B1 is the case of no scale economies, while Table B2is the case of strong scale economies in width and strongscale diseconomies in depth.

[89] Acknowledgments. This research was funded by the NationalCenter for Earth-surface Dynamics, a Science and Technology Center ofthe U.S. National Science Foundation (EAR-0203296). Helpful commentsby the reviewers and Editor are gratefully acknowledged.

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Table A1. Design: Single Stack of 8 m Culvertsa

Number of Row ofCulvert (8 m/row) Zb (m) Zt (m) W (m)

Water Discharge(m3/s)

Sand Discharge(m3/s)

Land BuiltYear 50 (km2) Nondominated

1 32 40 480 17,900 0.0635 67.2 H1 32 40 400 14,900 0.0530 57.9 H1 32 40 320 11,900 0.0424 48.2 H1 32 40 240 8940 0.0318 38.2 H1 32 40 200 7450 0.02648 33.0 H1 32 40 160 5960 0.0212 27.6 H1 32 40 120 4470 0.0159 22.1 H1 32 40 80 2980 0.0106 16.3 H1 32 40 40 1490 0.00530 10.3 N, H1 24 32 480 17,900 0.178 1551 24 32 400 14,900 0.148 133 H1 24 32 320 11,900 0.119 111 H1 24 32 240 8940 0.0891 88.21 24 32 200 7450 0.0742 76.0 H1 24 32 160 5960 0.0594 63.4 H1 24 32 120 4470 0.0445 50.11 24 32 80 2980 0.0297 36.1 H1 24 32 40 1490 0.0148 21.0 N, H1 16 24 480 17,900 0.337 2551 16 24 400 14,900 0.280 2211 16 24 320 11,900 0.224 1851 16 24 240 8940 0.168 1471 16 24 200 7450 0.140 1271 16 24 160 5960 0.112 1061 16 24 120 4470 0.0841 83.81 16 24 80 2980 0.0561 60.21 16 24 40 1490 0.0280 34.3 N, H1 12 20 480 17,900 0.458 3231 12 20 400 14,900 0.382 2801 12 20 320 11,900 0.305 2351 12 20 240 8940 0.229 1871 12 20 200 7450 0.191 1621 12 20 160 5960 0.153 1351 12 20 120 4470 0.114 1071 12 20 80 2980 0.0763 77.11 12 20 40 1490 0.0382 43.9 N, H1 8 16 480 17,900 0.644 4171 8 16 400 14,900 0.537 3631 8 16 320 11,900 0.430 3051 8 16 240 8940 0.322 2441 8 16 200 7450 0.268 211 H1 8 16 160 5960 0.215 1771 8 16 120 4470 0.161 1401 8 16 80 2980 0.107 1011 8 16 40 1490 0.0537 57.6 N, H1 4 12 480 17,900 1.00 576 N, H1 4 12 400 14,900 0.834 502 N, H1 4 12 320 11,900 0.667 423 N, H1 4 12 240 8940 0.500 339 N, H1 4 12 200 7450 0.417 294 N, H1 4 12 160 5960 0.334 247 N, H1 4 12 120 4470 0.250 197 N, H1 4 12 80 2980 0.167 143 N, H1 4 12 40 1490 0.0834 81.7 N, H

aSand discharge in m3/s can be converted to MT/yr by: 1 MT/yr¼ (m3/s) � (2650 kg/m3) � (1 T/1000 kg) � (86,400 s/day) � (365.25 day/yr) �(75.75 day/365.25 days) � (1 � 10�6 MT/T). Cost for each alternative can be calculated from Zb, Zt, and W using: Cost / (40� Zb) � W and assumedvalues of coefficients, i.e., N (no-scale economies): Cost / (40�Zb) �W, and H (high-scale effects): Cost / (40�Zb) 1.6 �W0.6.

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Table A2. Design: Two Stacks of 8 m Culverts

Number of Row ofCulvert (8 m/Row) Zb (m) Zt (m) W (m)

Water Discharge(m3/s)

Sand Discharge(m3/s)

Land BuiltYear 50 (km2) Nondominated

2 24 40 240 17,900 0.121 113 H2 24 40 200 14,900 0.101 97.6 H2 24 40 160 11,900 0.0806 81.4 H2 24 40 120 8940 0.0604 64.4 H2 24 40 80 5960 0.0403 46.2 H2 24 40 40 2980 0.0201 26.5 N, H2 16 32 240 17,900 0.257 2072 16 32 200 14,900 0.214 179 H2 16 32 160 11,900 0.172 150 H2 16 32 120 8940 0.129 1192 16 32 80 5960 0.0858 85.3 H2 16 32 40 2980 0.0429 48.5 N, H2 12 28 240 17,900 0.353 2642 12 28 200 14,900 0.294 229 H2 12 28 160 11,900 0.235 1922 12 28 120 8940 0.176 1532 12 28 80 5960 0.118 1102 12 28 40 2980 0.0588 62.6 N, H2 8 24 240 17,900 0.491 3402 8 24 200 14,900 0.409 2952 8 24 160 11,900 0.327 248 H2 8 24 120 8940 0.245 1972 8 24 80 5960 0.164 143 N, H2 8 24 40 2980 0.0818 81.5 N, H2 4 20 240 17,900 0.730 457 N, H2 4 20 200 14,900 0.608 398 N, H2 4 20 160 11,900 0.487 335 N, H2 4 20 120 8940 0.365 268 N, H2 4 20 80 5960 0.243 195 N, H2 4 20 40 2980 0.122 112 N, H

Table A3. Design: Three Stacks of 8 m Culverts

Number of Row ofCulvert (8 m/row) Zb (m) Zt (m) W (m)

Water Discharge(m3/s)

Sand Discharge(m3/s)

Land BuiltYear 50 (km2) Nondominated

3 16 40 160 17,900 0.193 165 H3 16 40 120 13,400 0.145 131 H3 16 40 80 8940 0.0964 94.0 H3 16 40 40 4470 0.0482 53.4 N, H3 12 36 160 17,900 0.275 2183 12 36 120 13,400 0.207 173 H3 12 36 80 8940 0.137 1253 12 36 40 4470 0.0687 71.2 N, H3 8 32 160 17,900 0.386 283 H3 8 32 120 13,400 0.290 226 N, H3 8 32 80 8940 0.193 164 N, H3 8 32 40 4470 0.0966 93.7 N, H3 4 28 160 17,900 0.569 380 N, H3 4 28 120 13,400 0.427 304 N, H3 4 28 80 8940 0.285 221 N, H3 4 28 40 4470 0.142 127 N, H

Table A4. Design: 4 m Culverts, Single Stack

Number of Row ofCulvert (8 m/row) Zb (m) Zt (m) W (m)

Water Discharge(m3/s)

Sand Discharge(m3/s)

Land BuiltYear 50 (km2) Nondominated

0.5 36 40 480 8430 0.0170 23.2 H0.5 36 40 400 7030 0.0141 20.2 H0.5 36 40 320 5620 0.0113 17.1 H0.5 36 40 240 4220 0.00848 13.9 H0.5 36 40 200 3510 0.00706 12.3 H0.5 36 40 160 2810 0.00565 10.7 H0.5 36 40 120 2110 0.00424 9.01 H0.5 36 40 80 1410 0.00283 7.30 N, H0.5 36 40 40 703 0.00141 5.56 N, H

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Total Individual Characteristics

Land-BuildingGoal (km2)

Land Built(km2)

NormalizedCost

Water(m3/s)

Number ofProjects

Number ofProjects Zb Zt W Water

LandBuilt

NormalizedCost

100 100 8.56% 12,646 (31.62%) 18 18 36 40 40 703 6 0.5%200 201 22.42% 14,304 (35.76%) 14 1 4 28 40 4468 127 16%

1 36 40 80 1405 7 0.7%12 36 40 40 703 6 0.5%

300 304 30.92% 13,404 (33.51%) 1 1 4 28 120 13,404 304 31%400 403 42.48% 15,596 (38.99%) 2 1 4 20 200 14,893 398 42%

1 36 40 40 703 6 0.5%500 510 58.00% 17,872 (44.68%) 2 1 4 20 200 14,893 398 42%

1 4 20 40 2979 112 16%600 607 77.78% 17,872 (44.68%) 2 1 4 12 240 8936 339 47%

1 4 20 120 8936 268 31%700 701 100.00% 17,872 (44.68%) 3 2 4 12 200 7447 294 42%

1 4 20 40 2979 112 16%

Table B1. Efficient Portfolios for No-Scale Economies (¼ 1, ¼ 1) Resulting From Minimizing Cost Subject to Land Target andWater Less Than 18,000 m3/sa

Total Individual Characteristics

Land-BuildingGoal (km2)

Land Built(km2)

NormalizedCost

Water(m3/s)

Number ofProjects

Number ofProjects Zb Zt W Water

LandBuilt

NormalizedCost

100 112 14% 2979 (7.45%) 1 1 4 20 40 2979 112 14%200 209 27% 7447 (18.62%) 2 1 8 24 40 2979 82 13%

1 4 28 40 4468 127 14%300 308 38% 13,404 (33.51%) 3 1 16 40 40 4468 53 10%

2 4 28 40 4468 127 14%400 403 49% 14,893 (37.23%) 4 1 24 32 40 1489 21 6%

3 4 28 40 4468 127 14%500 510 57% 17,872 (44.68%) 4 4 4 28 40 4468 127 14%600 609 81% 17,872 (44.68%) 6 1 16 24 40 1489 34 10%

4 4 20 40 2979 112 14%1 4 28 40 4468 127 14%

700 722 100% 17,872 (44.68%) 7 2 4 12 40 1489 82 14%5 4 20 40 2979 112 14%

aEquation to translate water to %: percentage of water use¼water used/40,000 (total amount of water) � 100%.

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