| Contextual Infrastructure Planning and Design |
| Minimum Cost Canals: How Low Can You Go? |
| By David C. Froehlich, Morrisville, North Carolina. 1-919-468-2131,
froehlich@pbworld.com |
|
F igure 1: Artificial canals convey water for navigation, crop,
irrigation, water supply, and drainage. Canals are usually connected
to natural bodies of water or other canals. A canal bringing
water for crop irrigation is shown in the photograph |
Canals are artificial waterways built for navigation, crop irrigation,
water supply and drainage. They are usually connected to natural bodies
of water or other canals (Figure 1). Ancient Phoenicians, Assyrians,
Sumerians and Egyptians constructed elaborate canal systems. A major
system of irrigation canals and ditches, for example, was built during
the reign of the Babylonian king, Hammurabi (1792-50 BC). Irrigation
was so vital to Mesopotamia that an ancient Babylonian curse was:
"May your canal be filled with sand!" During the 7th century
BC, the Assyrian king, Sennacherib, had an 80-km (48-mile)-long stone-lined
canal constructed to bring fresh water to the city of Nineveh. The
most spectacular canal of this period was probably Nahrawan. It was
120 m (400 feet) wide and 320 km (192 miles) long, and built to provide
a year-round navigation channel that used water obtained by damming
the unevenly flowing Tigris River.
Artificial canals can depart from natural river valleys, passing through
hills and watersheds and crossing over valleys and streams. Beds and
banks of canals can be formed from natural soils or lined with flexible
or rigid materials, such as loose rock or concrete. Linings are provided
to protect channel perimeters from erosion, increase flow efficiency
and reduce seepage.
As with most engineering projects, economic canal designs that convey
specified flow rates at the lowest possible costs are usually desired.
Achieving the least expensive configurations involves minimizing the
sum of land costs, earthwork costs, lining costs, and the cost of
water lost by evaporation and seepage. Trout (1982) presents a direct
solution for minimizing channel lining costs alone. Swamee et al.
(2000) include other costs as well while developing approximate empirical
relations for designing minimum cost canals. Because the problem is
difficult to solve analytically, the general-purpose optimizer called
Solver, which is contained in most spreadsheet programs, is used here
to obtain cross section dimensions that yield minimum cost canals.
The numerical solutions are exact for all practical purposes and can
be carried out quickly and easily using a commonly available computational
tool.
Figure 2: Artificial canals are often built with cross sections
of trapezoidal shape as shown. Lining thickness and freeboard
heights have been exaggerated for illustration purposes |
 |
Canal Cross
Sections
Canal cross sections can take on many shapes, but usually easily constructed
cross section shapes such as rectangles and trapezoids are built for
practical reasons. A typical trapezoidal cross section including linings
along the sides and bottom is shown in Figure 2.
Freeboard.
Canal side linings and canal banks are extended above normal water-surface
elevations to prevent overtopping and possible failure of the channel
from higher than expected water depths. Water depth fluctuations may
be caused by sedimentation, excess flows caused by storm runoff entering
through drain inlets, increased flow resistance due to deterioration
of channel bottoms and banks, waves produced by wind or by surges
that accompany sudden changes in flow rates, or by temporary improper
operation of a canal system.
Freeboard is the vertical
distance that canal banks or linings are extended above design water
surfaces to provide adequate margins of safety. Suggested freeboards
for canal banks, hard-surface and buried membrane linings and compacted
earth linings are provided by Aisenbrey et al. (1978, page 15) as
functions of canal flow rates.
Geometric Parameters.
Cross section flow area A and wetted perimeter P of a trapezoidal
canal cross section are calculated as follows:
(1) 
and
(2)
where B = bottom width of flow area, D = water depth, and z
= side-slope ratio (horizontal to vertical; z = 0 denotes
vertical sides and a rectangular cross section). Total section area
Ae needed for earthwork calculations is given by
(3)
where
(4)
is bottom width of the excavated cross section and of the bottom lining,
and
(5)
is the total height cross section, tb = thickness of bottom lining,
and f2 = canal bank freeboard or the vertical distance from canal
banks to the normal canal water surface.
Canal
Costs
Total cost of artificial canals considered here includes the costs
of land, earthwork, channel side and bottom linings, and water lost
by evaporation and seepage.
Land Cost. Rights-of-way
needed for construction of canals are usually of uniform width and
include provision for vehicle access along both sides. Access needs
do not depend on canal cross section dimension so their costs are
disregarded in this analysis. Only land containing the excavate canal
need be considered. Cost of land for a unit length of canal is then
given by
(6)
where µa = cost per unit area of land, which might include costs
of subsurface investigations, land preparations, and runoff and erosion
control measures in addition to the purchase cost, and
(7)
is the total topwidth of the canal cross section.
Earthwork
Cost. Excavation or fill needed to construct canal sections is
included in earthwork cost. Earthwork cost for a unit length of canal
is calculated as
(8)
where µe = earthwork cost per unit volume of excavation or fill.
Lining Cost. Costs of canal side and bottom linings per
unit length are calculated based on the volumes of material used to
construct them as follows:
(9)
where µb = cost of bottom lining per unit volume, tb = thickness
of bottom lining, µs = cost of side lining per unit volume,
ts = thickness of side lining, and f1 = side lining freeboard or the
vertical distance from the top of side linings to the normal canal
water surface.
Water Cost. Water is lost from canals
by seepage through the sides and bottom and by evaporation from the
water surface. Seepage rates from unlined canals can be extremely
large, and even lined channels never seem to eliminate water loss
through sides and bottoms. Measured seepage rates from lined canals
vary widely (Fipps 2000). The best concrete-lined channels may lose
about 8 mm/day of water through wetted boundary surfaces. Rate of
seepage per unit length of canal is calculated here simply as
(10)
where F = seepage rate per unit area. Evaporation depends on air temperature,
relative humidity or air above the water surface and wind speed. Fulford
and Sturm (1984) suggest formulas for estimating evaporation from
flowing open channels. Evaporation per unit length of canal is given
by
(11)
where T = width of the water surface, and E = evaporation rate per
unit surface area.
Assuming a very long canal life, the capitalized cost of water lost
per unit length of canal is given by
(12)
where µw = cost of a unit volume of water, r = annual discount
rate, and the leading factor converts loss rates from unit volumes
per second to unit volumes per year. Water costs vary widely throughout
the world, depending on supply, demand and water pricing policies.
Problem Constraints
Constraints are requirements on the values of selected problem variables.
For our optimization problem, flow rate Q is required to equal the
value given by Manning's formula (Chow 1959, page 98), that is
(13)
where ø = unit conversion constant (1.0 for SI units and 1.486
for U.S. Customary units), n = Manning's roughness coefficient,
and So = longitudinal channel bed slope. Bottom width of
the cross section B and the side-slope ratio z need to equal or exceed
zero (B = 0 yields a triangular cross section shape, and z
= 0 produces a rectangular cross section shape).
Average canal velocities V = Q/A may also be of concern. If water
travels too slowly, sediment carried by the flow can deposit and lead
to higher water-surface elevations and reduced capacities. On the
other hand, water moving at high speeds can erode beds and banks.
For water carrying no silt load, minimum velocity has little significance
except for its effect on plant growth. In general, minimum average
barrel velocities of 0.6 to 0.9 m/s (2 to 3 feet/sec) are suitable
when the percentages of silt-sized material present in channel flows
are small, and average velocities greater than 0.8 m/s (2.5 feet/sec)
will prevent growth of vegetation that might decrease flow-carrying
capacities of channels (Kennedy, 1895; Coleman 1923, page 70; "Minimum
velocities" 1942; Chow 1959, page 158; "Gravity sanitary
sewer design" 1982, page 105; "Design and construction"
1992, page 273). Maximum allowable velocities that prevent erosion
are usually based on the types of channel lining. Limitations might
also be imposed because of large superelevation in bends and high
degrees of wave action. Maximum permissible velocities for various
canals are suggested by Etcheverry (1915), Fortier and Scobey (1926),
Lane (1955), and Swamee et al. (2000).
Cross section topwidth, height, and side slopes may also be restricted
by site conditions or construction-related factors. Upper and lower
limits, therefore, may constrain channel dimensions.
Optimization
Problem
Finding the dimensions of the minimum cost canal having a trapezoidal
cross section with side and bottom linings can be stated as a constrained
optimization problem with a nonlinear objective function as follows:
Minimize (14)
subject to
(15)
and the bound constraints
(16)
where the superscripts L and U denote lower and upper limits, respectively.
With Q, n, So, f1, f2, tb, ts, F, E, r, µa, µe, µb,
µs, µw, and the lower and upper limits of z, H, Te, and
V specified for a channel, the cross section shape parameters B, z,
and H need to be adjusted (although if two of these values are known
the third can be calculated). Cross section and lining parameters,
and cost components depend on selected values of the decision variables
B, z, and H.
Figure 3: Solver parameters dialog box. Parameters include
the target cell (objective function), changing cells (decision
variables) and constraints. |
Spreadsheet Optimization
The general-purpose optimizer called Solver, which was developed by
Frontline Systems for small-scale linear, integer and nonlinear programming
problems, is included with every copy of Microsoft's Excel and Corel's
Quattro Pro spreadsheet programs. Froehlich (1999) describes how to
use Solver and applies it to optimize a storm sewer system. To summarize
briefly, Solver (see the dialog box in Figure 3) automatically tries
to maximize or minimize an objective function, which is just
a spreadsheet cell containing a formula, by modifying values used
in the function. Values modified by Solver, which are called changing
cells or decision variables, are simply other cells
in your spreadsheet containing numbers. Solver looks for input values
that satisfy a set of equations (involving "=" relations)
and inequalities (those involving "<=" or ">="
relations), that is, the constraints, and at the same time maximizes
or minimizes the objective function or target cell value.
Model. Canal cost per unit length given by (14) is
our objective function. We want it to be as small as possible. A spreadsheet
model using Solver was developed to calculate all
Figure 4: Specified parameters entered in the spreadsheet
column C include unit costs, contents values for the canal cross
section, and bound constraints on solution variable |
Figure 5: Values that change during the solution are contained
in spreadsheets column G. Included are the decision variables
(B, z, and D), cross section parameters, linig parameters and
costs. The objective function is in cel G30 (the target cell) |
cross section properties, flow rate, and component costs based on
specified unit costs and constant values, and upper and lower limits
of cross section dimensions and average velocities. Unit costs are
entered in spreadsheet cells C2 through C6 as shown in Figure 4. Constant
values specified for the problem are entered in cells C9 through C1.
Upper and lower bound constraints are given in cells C21 though C26.
Decision variables are B, z, and D. They are displayed in cells G2
through G4, and the objective function is contained in cell G29 as
shown in Figure 5. Flow cross section parameters, total cross section
parameters, lining parameters and component costs are also contained
in column G of the spreadsheet, as shown in Figure 5. Initial values
of B, z, and D that provide feasible solutions need to be entered
before running Solver. All other variables can be found given values
of these three parameters. Once started, Solver continues adjusting
values of B, z, and D until the lowest cost canal that satisfies all
of the constraints is found.
Reports. Besides adjusting spreadsheet cells and calculating
total cost, Solver creates three types of reports that are placed
on separate sheets in the workbook.
- Answer reports list the
target cell and adjustable cells with their original and final
values, constraints, and information about the constraints.
- Sensitivity reports provide
information about how sensitive the solution is to small changes
in the target cell formula and the constraints.
- Limits reports list the
target cell and the adjustable cells with their respective values,
lower and upper limits, and target values.
Example Application
Cross section dimensions of a lined open channel are found to show
how Solver can be used to minimize the total cost of an irrigation
canal. The canal is to be lined with concrete on the bottom and sides,
and needs to convey a design flow rate Q = 12 m3/s on a longitudinal
bed slope So = 0.0016. Manning's roughness coefficient for the float-finished
concrete lining n = 0.015 (Chow 1959, page 111). The canal passes
through a stratum of ordinary soil for which tb = 0.30 m, ts = 0.15
m, µa = $1/m2, µe = $10/m3, and µb = µs =
$200/m3. The unit cost of water µw = $1.00/m3, and the annual
discount rate r = 5%. Seepage from the new, well-maintained concrete
lining is estimated to be 8 mm/day (9.3x10-8 m/s), and climatic conditions
produce average evaporation E = 6 mm/day (6.9x10-8 m/s). Freeboard
heights (Aisenbrey et al. 1983, page 15) for the design flow rate
are as follows: f1 = 0.35 m, f2 = 0.85 m. Width, depth, and side slope
of the canal cross section are not constrained, so we assign HL =
TeL = zL0 (physical lower limits of each of the variables), and HU
= TeU = zU 1x106 (an arbitrarily large value that will not be approached
during the optimization process). Constraint bounds are placed on
average channel velocity so that 0.6 m/s = V = 2.5 m/s.
Specifications (that is, fixed values and constraints) for the problem
are shown in the section of the spreadsheet displayed in Figure 4.
The section of the spreadsheet containing variable quantities including
decision variables, cross section and lining parameters, cost components,
and the objective function (that is, total cost per unit length of
canal) are shown in Figure 5. Solver's Answer report is presented
in Figure 6 on the following page.
Figure 6: Solver's Answer report for
the example problem showing initial and final values of the
objective function and decision variables, and the status of
all equality and bound constraints |
Solver finds that a trapezoidal cross section with B = 1.48 m, z =
0.40, and D = 2.17 m provides the least expensive canal costing $868.28
per meter length. Total height and topwidth of the canal section are
3.02 m and 3.90 m, respectively. Typical trapezoidal cross sections
of concrete-lined irrigation canals in the western U.S. have bottom
width to water depth ratios (that is, values of B/D) ranging from
1 to 2, and side-slope ratios ranging from 1.25 to 1.5 (Houk 1956).
Our minimum-cost solution yields B/D = 1.48/ 2.17 = 0.68, a value
far less than usually found in existing lined canals, and much steeper
sides. If we set zL = zU = 1.5 (that is, if we fix the side slope
ratio z = 1.5), Solver finds that a cross section of triangular shape
(that is, with B = 0 m) having a normal water depth D = 1.88 m produces
the minimum cost canal ($1,046/m length). Based on our solutions for
cross section dimensions yielding the least expensive canal, it seems
that concrete-lined irrigation channels in the U.S. might not have
always been designed in the most economical ways. Different unit costs,
especially of land and water, can lead to much different cross section
dimensions. Variability of costs and other parameters, such as seepage,
evaporation and lining thicknesses, can be studied easily using the
spreadsheet optimizer.
Summary and
Conclusions
The spreadsheet optimizer Solver is used here to decide on cross section
dimensions that produce the minimum cost canal design. Canals may
be lined or unlined, and restricted by limits on cross section dimensions
and average channel velocities. Costs include those related to land
acquisition, earthwork, lining construction and water loss from seepage
and evaporation. Once the optimization model is created in your spreadsheet,
Solver is easy to run. Effects of changes to problem parameters and
bound constraints can be evaluated quickly and easily to see the influence
of those values on the objective function, that is, the total canal
cost. |
|
Dave Froehlich, Senior Supervising Water Resources
Engineer, solves problems related to river mechanics, sediment transport
and hydraulic structures for PB; he is based in Raleigh, North Carolina.
Dave has developed several computational tools for the Federal Highway
Administration to assess stream flow and riverbed scour at bridges.
References
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N.Y.
Coleman, G. S. (1923). Hydraulics applied to sewer design. Crosby
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"Design and construction
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