ResearchArticle ... · components of the total cost of a pipe drain project. ... (unplasticized...

Research ArticleThe Study of Subsurface Land Drainage Optimal Design Model

Ahmad Bakour 1 Zhanyu Zhang 1 Chengxin Zheng 2 Mohamed A ALsakran 3

and Mohamad Bakir 4

1College of Agricultural Science and Engineering Hohai University Nanjing 210098 China2College of Water Conservancy and Hydropower Engineering Hohai University Nanjing 210098 China3College of Civil and Transportation Engineering Hohai University Nanjing 210098 China4College of Hydrology and Water Resources Hohai University Nanjing 210098 China

Correspondence should be addressed to Zhanyu Zhang zhanyuhhueducn

Received 29 September 2020 Accepted 4 June 2021 Published 16 June 2021

Academic Editor Tianwei Zhang

Copyright copy 2021AhmadBakour et alis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper focused on choosing the best design of subsurface land drainage systems in semiarid areas e study presented threedifferent soil layers with different hydraulic conductivity and permeability all layers are below the drain level and the permeabilityis increasing with depth Amathematical model was formulated for the horizontal and vertical drainage optimal designe resultwas a nonlinear optimization problem with nonlinear constraints which required numerical methods for its solution epurpose of the mathematical model is to find the best values of pipes and tubewells spacing groundwater table drawdown andpumps operating hours which leads to a minimum total cost of the subsurface drainage design A computer code was developed inMATLAB environment and applied to the case study Results show that the vertical drainage was economically better for the casestudy drainage network design And the main factor affecting the mathematical model for both pipe and well drainage was thedistance between pipes and tubewells In addition considering the lifespan of vertical drainage project the optimal design involvesthe minimum possible duration of pumping stations It is hoped that the proposed optimal mathematical model will present adesign methodology by which the costs of all alternative designs can be compared so that the least-cost design is selected

1 Introduction

Subsurface drainage is widely used worldwide to removeexcess water found below the earthrsquos surface [1] Whilesurface drainage removes excess water from the soil surfacebefore it enters the crop root zone subsurface drainagedecreases the groundwater level and provides a better en-vironment for the crop growth [2] Agricultural lands af-fected by salinity and high water levels generally requiresubsurface drainage [3] Two methods of subsurfacedrainage are commonly used which are horizontal pipedrainage and vertical well drainage Subsurface pipedrainage is a group of lateral pipes spread below the groundsurface with specific depths diameters and spacing betweenthem and connected with collector pipes that draw thesurplus water out of the study area Vertical drainage is agroup of vertical wells with specific diameters depths andspacing between themese wells are equipped with pumps

that withdraw water so the water level will decrease in thesurrounding area Land drainage helps to achieve waterbalance in the soil prevent its salinization and createfavourable conditions for plant growth Around the worldagricultural drainage plays a significant role in protectinginvestments in irrigation projects and agricultural produc-tion It helps also in preserving soil resources and foodproduction by improving land productivity and crop yieldsespecially in poorly drained soils [4] Most studies thatinvolve subsurface drainage were about the design of thedrainage network and the task is to find the best design orthe optimal design e most important strategy in de-signing the subsurface drainage is choosing proper values forthe spacing and depth of drains to minimize the total cost[5] And a lot of researchers have studied the cost analysis ofsubsurface drainage systems for different types [6ndash11]Cimorelli et al [12] have worked on the surface drainagenetwork optimal design and introduced a novel procedure

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 8827300 11 pageshttpsdoiorg10115520218827300

for the best surface drainage network design Amethodologywas presented by Chahar and Vadodaria [8] for deter-mining the optimal spacing of ditches fully penetrating intoan isotropic and homogeneous porous medium over animpervious layer Another methodology was introduced byMoradi-Jalal et al [13] for the optimal operation and designof pumping stations based on solving a nonlinear large-scaleprogramming problem And a new management model waspresented by Moradi-Jalal et al [14] for the optimal op-eration and design of water distribution systems An ob-jective function was constructed by Sharma and Swamee[15] for the cost structure of a pipe network system eoptimization model described by Bennett and Mays [16] wasbased upon dynamic programming for nonserial systemsthat determines the minimum cost of a drainage channelsystem and the detention for a watershed A nonlinearsquare fitting routine was used by Wall and Miller [17] todetermine the effective values of conductivity and moisturecharacteristics of the model of soil water drainage And asimulation model of the optimal drainage network waspresented by Howard [18] in which channels shift tominimize the total stream power within the network

is paper presents a novel strategy for the best design ofsubsurface horizontal and vertical drainage in an area ofdifferent saturated soil layers with different hydraulic con-ductivity and permeability and the permeability is in-creasing with depth With the use of modern optimizationalgorithms we can find the suitable values of groundwatertable drawdown and pipestubewells spacing that lead to theminimum cost of the total subsurface drainage system

2 Methodology

e least-cost design is that satisfying all design constraintswith the minimum total cost e objective function for pipeand well drainage was determined by considering all costcomponents that affect the drainage network design enconstraints were formulated depending on the hydraulicstudy of the study area e result was a nonlinear objectivefunction with nonlinear constraints A survey of modernoptimization algorisms was conducted to find the onesuitable for the solution of the formulated optimizationproblem It was found that the interior-point optimizationalgorithm was adapted to the problem and produced sat-isfactory results Two computer codes for both horizontaland vertical subsurface drainage were developed in MAT-LAB environment in order to derive the optimal solution forboth types of drainage systems e solutions then werecompered to know how the lifespan and type of the projectwill affect the total cost of the network design

3 Pipe Drainage Costs

e major cost components of pipe drainage system aredrainage materials installation and operation andmaintenance

31 DrainageMaterials e costs of drainage materials anddrainage pipe installation work and structures are major

components of the total cost of a pipe drain projectDrainage materials include collector and lateral pipes filterspump set and pump house outlet structures and manholesand outlet pitching

Collector and lateral pipes the collector pipes have atransmission function to carry water to the outlet under thegravitational flow so the UPVC (unplasticized poly vinylchloride) corrugated nonperforated pipe is used for thecollectors e pipe diameter of the collectors is chosendepending on the expected flow Single Wall PerforatedFlexible Corrugated UPVC Pipes (outer diameters of80ndash355mm) are widely used for lateral pipes

Artificial filtersenvelopes the filter or envelope materialaround the pipes plays an important role in preventing thefine particles of the soil from entering into the pipes with theflow Nowadays it is preferred to use artificial filters ratherthan gravel filters e artificial filters are cheaper in cost ofbetter quality and easier to handle and transport than gravelfilters For medium and light soil it is preferred to usenonwoven polypropylene fibers with a thickness of morethan 25mm and an opening size of more than 300 micronsand a needle punched geotextile nonwoven fabrics are usedfor heavy soil Perforated pipes precoated with industrialfilter are more preferred than locally coated pipes for qualityinstallation

Pump set and structures pump set and structure consistsof diesel pump set and pump house manholes (junctionbox) and outlet structures Other cross drain structures mayalso be required in larger projects but these are avoided forsimplicity e outlet structure and manholes are precastReinforced Cement Concrete (RCC) pipes of differentlengths and diameters depending on the site conditions Inthe case of pump outlets 900mm diameter manholes and1200mm outlet structures are used where 900mm diametermanholes are used in the gravity outlets e bottom edge ofthe RCC pipe for outlet structures and manholes is closed byfixed plate at the bottom Plastic coated iron bars are pro-vided in the walls of outlet structures and manholes to helpin inspection and cleaning In the case of gravity outletsinstead of RCC pipes plastic manholes and outlet pipes haverecently been used Plastic manholes and outlet pipes arerelatively expensive but they are easy to carry handle andinstall at the project sites [19] A small stone structure isconstructed at the outlet to protect the pipe end fromcollapsing under the conditions of gravity outlet A rodentguard is provided to prevent the pipes from getting stuck anddamaged by rodents that may enter the lateral pipes

32 Installation Installing the drainage system is thefunction of the installation unit according to the specifi-cations and design Supervisors must specify that the in-stallation (including drainage equipment) is carried outstrictly in accordance with specifications and design Var-ious drainage machines such as hydraulic excavatorstractor-mounted trenchers self-propelled trenchers withlaser automatic control and self-propelled machines with LorV plough and laser control are used to install collector andlateral pipes [20 21] Currently laser control self-propelled

2 Mathematical Problems in Engineering

trenchers are used in large-scale quality of subsurfacedrainage pipes installations Other auxiliary machineries likebulldozers excavators tractors with trailers backhoe etcare also used for the movement of manpower drainagematerials and installation of outlets and manholes Indi-vidual farmers can use tractor mounted trencher in smallareas (1ndash5 ha) to install subsurface drainage pipes In suchcases it may be desirable to use a laser control device with atractor-mounted trencher to achieve suitable slope forcollector and lateral pipes

33 Operation and Maintenance e popular belief thatsubsurface drainage does not require any maintenance andoperation is untenable In the case of pump outlets pumpoperation is required for at least the first few years of in-stallation e maintenance of subsurface drainage systemsmainly involves removing sediment from outlets manholesand pipes also repairing or replacing the damaged outletsmanholes and pipes [20] In controlled drainage systemsoperations may also include closing and opening gates toreuse drained water to irrigate crops

34PipeDrainageCostEquation emost cost componentsthat affect the subsurface pipe drainage design can be de-termined according to the total costs as shown in the fol-lowing relationship

zi Wi +1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857DlowastUi (1)

where i is the number of the choice (minus) zi is the total costsfor the choice i ($) E0 is the interest rate () Wi is theconstruction costs for the choice i ($)D is the lifespan of thedrainage project (years) and Ui is the annual investmentcosts for the choice i ($year)

e construction costs are calculated by using the fol-lowing equation

Wi 1113944 li lowast Acut lowastCcut + 05lowast ICcut lowastAcut lowast hcut + Cp1113872 1113873 + nm lowast hcut lowastCm (2)

where 1113936 li is the total length of all drainage pipes (m) Acut isthe cross-sectional area of excavation (m2) Ccut is the unitcost of excavation ($m3) ICcut is the increase in unit cost ofearthwork per unit depth of excavation ($m3m) hcut is thedepth of excavated ditch (m) Cp is the unit cost of drainagepipes and filters ($m) Cm is the unit cost of manholes perdepth ($m) and nm is the total number of manholes (minus)which can be determined by

nm Atot

Elowast S (3)

where Atot is the total area of the study area (m2) E is pipespacing for subsurface pipe drainage network (m) and S isthe distance between manholes along the drainage pipe (m)

e cross-sectional area of excavation (Acut) is a trap-ezoid section as shown in Figure 1 and it can be calculated by

Acut b + hcut lowast tan empty( 1113857lowast hcut (4)

where b is the bottom width of excavation hcut is the totaldepth of excavation and empty is the lateral slop angle

Also total length of all drainage pipes (1113936 li) can bereplaced by

1113944 li ni lowast l (5)

where l is the mean length of all lateral drainage pipes (m)and ni is the total number of drainage pipes (minus) and it can bedetermined by using the following equation

ni Atot

Ai

(6)

where Ai is the mean area served by drainage pipes (m2) andit can also be determined by using the following equation

Ai Elowast l (7)

So the construction costs can be written as

Wi Atot

Elowast llowast llowast b + hcut lowast tan empty( 1113857lowast hcut( 1113857lowastCcut + 05 lowast ICcut lowast b + hcut lowast tan empty( 1113857lowast hcut( 1113857lowast hcut + Cp1113872 1113873 +

Atot

Elowast Slowast hcut lowast cm

(8)

b

hcut

ϕ

Figure 1 Cross section of excavation

Mathematical Problems in Engineering 3

Moreover the annual investment costs are

Ui PWi + Uti (9)

where P is the rate of depreciation and it is taken as 15-16years Ut

i is annual cost of maintenance and service for thedrainage network ($) And it can be calculated by using thefollowing relationship

Uti nm lowastCs (10)

where Cs is the annual cost of maintenance and service foreach manhole ($)

e objective is to design the least-cost pipe drainagenetwork us the objective function can be stated as

Zi 1 +1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857DlowastP⎛⎝ ⎞⎠

Atot

Elowast b + hcut lowast tan empty( 1113857lowast hcut( 1113857lowastCcut + 05 lowast ICcut lowast b + hcut lowast tan empty( 1113857lowast hcut( 1113857lowast hcut + Cp1113872 11138731113890

+Atot

Elowast Slowast hcut lowast cm1113891 +

1 + E0( 1113857D

minus 1E0 1 + E0( 1113857

Dlowast

Atot

Elowast SlowastCs

(11)

4 Well Drainage Costs

It is recommended to establish a proper strategy for verticaldrainage system design and it should be connected with theeconomic factors For example we can choose large num-bers of tubewells with a small amount of discharge and aslight decrease in the groundwater level from each tubewellor we can choose small numbers of tubewells with a largeramount of discharge and a larger decrease in the ground-water level from each tubewell and a larger spacing betweenthe tubewellsere are a lot of choices and these choices arecontrolled by

(a) Tubewell depth(b) Tubewell spacing(c) Tubewell discharge(d) e decrease amount of the groundwater level

And the most suitable choice can be determinedaccording to the total costs as stated in (1)

e construction costs for vertical drainage project arecalculated by using the following equation

Wi ni lowast lH lowastC1 (12)

where ni is the total number of tubewells lH is the total depthof the tubewell (m) and C1 is the construction cost of eachtubewell per one meter depth ($m)


Ui PWi + Uti + U

Ni (13)

where Uti is the annual cost of maintenance and service for

the tubewells ($) And it can be calculated by using thefollowing relationship

Uti ni lowastC2 (14)

where C2 is the annual cost of maintenance and service foreach tubewell ($) and UN

i is annual cost of electricity for thewithdrawal of water from each well ($) And it is given by

UNi

9813600 η0

Qi lowast hi lowast tlowastC3 (15)

where Qi is the discharge from each tubewell (m3day) hi isthe pumping depth (m) t is the pumps operating hours inthe year (day) C3 is the cost of kilowatt hours of electricity($kwh) and η0 is the pump efficiency (minus)

So the objective function for the design of verticaldrainage tubewells can be stated as

zi ni 1 +1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857DlowastP⎛⎝ ⎞⎠lowast lH lowastC1 +

1 + E0( 1113857D

minus 1E0 1 + E0( 1113857

D⎛⎝ ⎞⎠lowast C2 +

9813600 η0

lowastQi lowast hi lowast tlowastC31113888 1113889⎡⎢⎢⎣ ⎤⎥⎥⎦ (16)

5 Case Study

51 Case Study Description A total area of 500 hectaresis attended to apply an irrigation network to meet thecrops need and help in washing the salinity that comesfrom underground water So a drainage network is

needed also along with the irrigation network to serve inreleasing exceed salty water out of the study area estudy area is located in Syria as shown in Figure 2 enet area of cultivation is 405 hectares About 15 ofthe net area is covered by summer vegetables and wheatoccupies the largest proportion among the crops list


which is about 443 followed by cotton (35) andbarley (127)

Table 1 gives the general climate and crops indicators forthe region

e study area contains three different soil layers(h1 7m h2 12m and h3 15m) with different hy-draulic conductivity (k1 08 mday k2 32mday andk3 112mday) and there is a semi-impermeable layerunder the third layer with thickness of h4 2m and hy-draulic conductivity of k4 001mday as shown inFigure 3

e groundwater table is about 3m below the groundsurface the gradient of the groundwater surface at the be-ginning of the study area is J1 0006 and the gradient ofthe groundwater surface at the end of the study area isJ2 0001 e general gradient of the study area is fromnorth to the south so the length of groundwater feeding lineis L 2430m leakage from irrigation network isφ 6820789 m3hayear and the increasing amount ofirrigation discharge to avoid salinity problems isg 627276m3hayear

Unit cost of excavation is Ccut 50 (RMBm3) increasein unit cost of earthwork per unit depth of excavation isICcut 3 (RMBm3m) unit cost of drainage pipes and filtersis Cp 10 (RMBm) unit cost of manholes per depth isCm 200 (RMBm) annual cost of maintenance and servicefor each manhole is Cs 10 (RMB) empty 078 and distancebetween manholes along the drainage pipe is S 100mConstruction cost of each tubewell per one meter depth isC1 860 (RMBm) annual cost of maintenance and servicefor each tubewell is C2 80 (RMByear) the cost of kilowatthours of electricity is C3 052 (RMBkwmiddoth) pump efficiencyis η0 078 specific given factor is micro 006 rate of de-preciation is P 115 and interest rate is E0 8 edrainage depth is m 1m

52 Pipe Drainage Spacing Formulation We can determinethe pipe spacing for subsurface pipe drainage network whenwe have three different layers and the permeability is in-creasing with depth by using the following formulations[22]

E 4

f2

+ TH

2qc

1113971

minus f⎛⎝ ⎞⎠ (17)

where

T 1113944 Ki hi( 1113857

f β1β2 hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

(18)

As σ σ1 and σt can be calculated by using the followingformulations

σ 0366 Logh

2πr sin((H + r)2h)

σ1 0366 Loghlowast1

2πr sin (H + r)2hlowast1( 1113857

σt 0366 Loght

2πr sin (H + r)2ht( 1113857

(19)

where H is the hydraulic head (m) which is the water tableheight above the drainage pipe at the midpoint between thedrainage pipes as seen in Figure 4 T is the weightedtransmissivity for all layers contributing to the flow (m2day) qc is the drainage unit discharge that must be releasedby drainage pipes (mday) Ki is the hydraulic conductivityfor each layer (mday) hi is the thickness of each saturated

N

Figure 2 e location and topography of the study area


layer (m) h is the thickness of all saturated layers (m) r is theradius of drainage pipe (m) and hlowast1 and ht (m) can bedefined as

hlowast1 h1 minus m minus

H

2

ht hlowast1 + h2

(20)

where m is the drainage depth (m) which is the minimumdepth required by each plant for better productivity

And we can obtain β1 β2 and βt values from Figure 5after calculating the factors ψ λ and ε as seen in Table 2

e drainage unit discharge (qc) that must be released bydrainage pipes can be obtained by studying the water balancein the study area

53 Hydraulic Study of Drainage Pipes For the hydraulicstudy of subsurface drainage pipes we can use Manningformula which is as follows

3m

4m

12m

15m

2m

K1

K2

K3

K4

Ground waterinflow

Surfaceinflow

Surfaceoutflow

J1

J2

Ground wateroutflow

Rainfall

Ground water surface

Irrigationwater Leakage from

irrigationchannels

Evatranspiration

Deep groundwater flow

Figure 3 Hydrogeology of the study area

qc

h1

h2

h3

h4

hp

ht

E

Hm

Figure 4 Case study pipe drainage geometry

Table 1 General climate and crops indicators for the study area

Factor Month 1 2 3 4 5 6 7 8 9 10 11 12 e averageannual

Temperature 73 91 125 164 21 259 279 281 245 191 134 86 178

Rainfall (mm) 187 141 133 86 32 07 0 02 16 86 85 222 997Monthly rate 121 916 849 559 212 48 0 19 106 549 549 1456 649

Evaporation from freewater surface (mmday)

Lambert 12 17 24 26 57 79 105 95 65 38 14 11 48Ivanov 11 16 25 39 56 84 93 87 73 45 25 12 47

Evatranspiration (mmday) 07 18 25 44 66 84 89 89 61 32 2 07 456

Crop needs (mmmonth)

Wheat andbarley 1519 3528 5425 924 0 0 0 0 0 0 42 1519 2125

Cotton 0 3528 5425 924 14322 1764 19313 19313 0 0 0 0 7398Summervegetables 0 3528 5425 924 14322 1764 19313 0 0 0 0 0 5789


Q 1nlowastR

(23) lowastAlowastI

radic (21)

where Q is the discharge that must be drained by thedrainage pipes (m3sec) n is the roughness coefficient (minus) Ris the hydraulic radius (m) A is the water cross section area(m2) and I is the hydraulic gradient

For better calculation of drainage pipe diameter we canconsider that the pipe is full of water but we have to choose apipe with an actual diameter greater than the calculated onein order to guarantee the free surface flow inside thedrainage pipe

e calculated velocity inside the drainage pipes must bebetween these limits

1geVge 015msec (22)

And the critical pipe diameter must achieve the fol-lowing formula

dφ ge dcr 0262lowastqlowast

Km

(23)

where dφ is the drainage ditch width (m) dcr is the criticalpipe diameter (m) Km is the weighted hydraulic conduc-tivity for all layers contributing to the flow (mday) and qlowast isthe drainage unit discharge (m3daym) which can becalculated by using the following equation

qlowast

qc lowastElowast 1 (24)

54 Pipe Drainage Optimization Problem e optimizationproblem for the pipe drainage design can be stated asfollows

Minimize

Zi 1 +1 + E0( 1113857

Dminus 1

E 1 + E0( 1113857D

0

lowastP⎛⎝ ⎞⎠lowastAtot

Elowast b + hcut lowast tan empty( 1113857lowast hcut( 1113857lowastCcut + 05 lowast ICcut lowast b + hcut lowast tan empty( 1113857lowast hcut( 1113857(1113890

lowast hcut + Cp1113873 +Atot

Elowast Slowast hcut lowast cm1113891

+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

Atot

Elowast SlowastCs

(25)

08

09

10

07

06

ε = 00250510

2030ε = 50ψ gt = 01

β

λ

0 02 04 06 08 09 094 098 10

(a)

08

09

10

ε = 503020

1005025

ψ = 001

ε = 0β

λ

0 02 04 06 08 09 094 098 10

(b)

08

09

10β

ψ lt = 0001 λε = 50

30201005025ε = 0

0 02 04 06 08 09 094 098 10

(c)

Figure 5 Determination of βp β3 and β4 values [22]

Table 2 Calculation of ψ λ and ε factors for the values of β1 β2and βt [22]

β1 β2 βt

ψ (rhlowast1 ) (rhlowast1 ) (rhlowast1 )

λ (k3 minus k2k3 + k2) (k2 minus k1k2 + k1) (k2 minus k1k2 + k1)

ε (h3ht) (hphlowast1 ) (h2hlowast1 )


Subject to

E 4

f2

+ TH

2qc

1113971

minus f⎛⎝ ⎞⎠

2rge 0262lowastqlowastE

Km

H E gt 0

(26)

in which

f β1β2hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

σ 0366 Logh

2πr sin((2m + r)2h)


2πr sin (2m + r)2hlowast1( 1113857

σt 0366 Loght

2πr sin (2m + r)2ht( 1113857

(27)

55WellDrainageSpacingFormulation According to SovietScience Encyclopedia for calculating and design of drainagenetworks and land reclamation we can determine thetubewells spacing when the permeability is increasing withdepth by using the following formulation [22]

hsw hc +qc1 B

2

Tρ + fc( 1113857 (28)

where

H hsw minus hc

H qc1 B

2

Tρ + fc( 1113857

ρ 0336 middot logB

πrc

fc β3β4k1

k3σc + βp

k2 minus k1

k3σcp +

k3 minus k2

k3σc3

(29)

When the well is not reaching the impermeable layer(m2ne 0) then

σ 12π

1 minus x

xln07 lk

rc

+ ln1x

+Δε2

1113888 1113889 (30)

where

x lk

hi

(31)

For calculating σc we put (hi h hlowast1 + h2 + h3) forcalculating σcp we put (hi hp h2 + h3) and for calcu-lating σc3 we put (hi h3) as described in Figure 6

We can obtain the Δε values from Table 3And we can obtain βp β3 and β4 values from Figure 5

after calculating the factors λ ψ and ε as seen in Table 4In Table 4 rc is the radius of vertical well (m) Lk is the

length of the filter (m) B is the spacing between vertical wells(m) hsw is the maximum thickness of saturated layers (m) hcis the water depth inside the tubewell above the impermeablelayer (m) and qc1 is the drainage unit discharge that must bereleased by investment wells and it can be calculated byusing the following formula

qc1 p1 + p2

p1 φ + g

p2 104k4Δhh4

t

Δh p1

k1h1 minus m( 1113857

(32)

where p1 is the discharge that comes from surface water (mday) p2 is the discharge that comes from groundwater (mday) φ is the leakage from irrigation network (mday) and g

is the increasing amount of irrigation discharge to avoidsalinity problems (mday)

hchp

hth

h2

h3

B

hswLk

m2

H

qc2rc

m

Figure 6 Case study vertical drainage geometry

Table 3 Δε values according to the value (m + (Lk2))hi [22]

(m + (Lk2))hi 01 015 02 025 03 04 05

Δε 233 107 049 017 minus001 minus019 minus022

Table 4 Calculation of ψ λ and ε factors for the values of β3 β4and βp [22]

β3 β4 βpΨ (rhlowast1 ) (rhlowast1 ) (rhlowast1 )

Λ (k2 minus k1k1 + k2) (k3 minus k2k2 + k3) (k3 minus k2k2 + k3)

Ε (hlowast1 hp) (hth3) (h2h3)


56 Surrounding Wells Spacing Formula Protection verticalwells are placed at the edge of the study area along thefeeding line L to protect the study area from the upcominggroundwater along the feeding line is means that thesewells will act as an investment and protection role the deepgroundwater component coming to the study area throughthe fixed-length L and for a slide of 1m width is given by thefollowing relationship

G kmh J1 minus J2( 1113857

km 1113936 ki middot hi

h

(33)

where G is the groundwater component (m3mday) thatgathers in the study area during the day J1 is the gradient ofthe groundwater surface at the beginning of the study area(minus) and J2 is the gradient of the groundwater surface at theend of the study area (minus)

e spacing between surrounding wells can be calculatedby using the following formula

Bor

G2

4q2c1

+ B2

11139741113972

minusG

2 qc1 (34)

57 Determining the Number of Drainage Wells e totaldischarge that has to be released by all drainage wells can bedetermined by

Qtot Gyr lowast l + qc1 lowastF (35)

e discharge of each drainage well is

Qi qc1 lowastB2 (36)

us the total number of drainage wells is

n Qtot

Qi

(37)

e number of surrounding wells can be obtained by

nor L

Bor (38)

us the number of investment wells is

ni n minus nor (39)

58 Pump Operating Hours e duration of pump oper-ating required to maintain a favourable drainage depth isgiven by

tHc μπR

2H

2Qi

(40)

where μ is a specific given factor (minus) and R is the radius ofinfluence of the wells and it can be calculated according tothe distribution of the wells as follows

(i) If the tubewells are placed in a rectangular patternR 0565lowastB

(ii) If the tubewells are placed in a triangular patternR 0526lowastB

In our study we will choose a rectangular pattern Someresearchers suggest operating the pumps only during theweeding period but others prefer to operate the pumps incertain hours every day

59 Well Drainage Optimization Problem e optimizationproblem for the well drainage design can be stated as follows

Minimize

Zi Qtot1 + 1 + E0( 1113857

Dminus 1E0 1 + E0( 1113857

D1113872 1113873lowast P1113872 1113873lowast lH lowastC1 + 1 + E0( 1113857

Dminus 1E0 1 + E0( 1113857

D1113872 1113873lowastC2

qc1 lowastB2

⎡⎢⎣

+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

9813600 η0

lowast H + m)lowast tlowast C3( 1113859

(41)

Subject to

H qc1 lowast B

2

T0336 log

B

π rc

+ fc1113888 1113889

t μπ05652H

2 qc1

(42)

6 Results and Discussion

Table 5 shows the general parameters calculated for bothpipe and well drainage design

By applying these values on the computer code inMATLAB environment we derived the optimal solution forthe pipe drainage design for a range of lifespan as seen inTable 6

And the optimal solution for the vertical drainage designfor a range of lifespan is shown in Table 7

As we can see for horizontal and vertical drainagenetworks it is better to choose large distance between thelateral pipes and wells ese distances can be calculated byapplying the optimization model on the study area And forthe case study described above the vertical drainage will be abetter solution as subsurface drainage design for the whole


project lifespan e cost for operating vertical drainagepumps plays an important role in determining the optimaldesign when considering the lifespan of the project

7 Conclusions

In order to formulate an optimization problem for thedesign of subsurface drainage systems cost equations havebeen introduced for both horizontal and vertical drainagee cost equations contained the most cost components thataffect the subsurface drainage networks design en theoptimization problem constraints were derived from thehydraulic study of the case study e case study containsthree different soil layers with different hydraulic conduc-tivity and permeability and the permeability is increasingwith depth A mathematical model was formulated for thehorizontal and vertical drainage optimal design in the casestudye result was a nonlinear optimization problem withnonlinear constraints which required numerical methodsfor its solution A survey of modern optimization algorismswas conducted to find the one suitable for the solution of theformulated problem It was found that the interior-pointoptimization algorithm was adapted to the problem andproduced satisfactory results e results show that theproposed optimal mathematical model for both horizontaland vertical drainage networks was affected mostly by thedistance between pipes and wells and the optimal solutioninvolved the maximum possible values of pipes and tube-wells spacing Also for this case study the model gave alower cost for the designing of tubewells network comparedwith pipe network And the total cost for the verticaldrainage design involved minimum duration of pump op-eration when considering the lifespan of the subsurfacedrainage project e study has shown that the pipes and

tubewells spacing and the groundwater table drawdowncannot be selected randomly if we put the economic factor inconsideration Traditional pipes and tubewells design maylead to high costs compared with the optimal design It ishoped that the proposed optimal mathematical model willpresent a design methodology by which the costs of allalternative designs can be compared so that the least-costdesign is selected

Data Availability

e data that support the findings of this study are availablefrom the corresponding author upon reasonable request

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

e authors would like to thank Hohai University forgranting the scholarship which made the research possiblelab mates Genxiang Feng and Wang Ce for their sugges-tions and help friends Wael Alhasan and Saeed Assani fortheir big support and help is research was funded by theNational Natural Science Foundation of China under grantnumber 51879071

References

[1] C D Kennedy C Bataille Z Liu et al ldquoDynamics of nitrateand chloride during storm events in agricultural catchmentswith different subsurface drainage intensity (Indiana USA)rdquoJournal of Hydrology vol 466-467 pp 1ndash10 2012

Table 7 Well drainage optimal solution for a range of lifespan

LifespanD (years)

Hydraulichead H (m)

Duration ofpumpingoperation t

(days)

Wellspacing B

(m)

Total cost Z(RMB)

edischarge ofeach well Qi(m3day)

Totalnumberof wells n

Surroundingwells spacing

Bor (m)

Number ofsurroundingwells nor

Number ofinvestmentwells ni

1 175848 1849211 8810597 29559e+ 05 22208976 75 7167423 34 4110 99645 1047862 6737206 70352e+ 05 12986063 128 5150194 47 8150 87810 923413 6345266 10038e+ 06 11519073 145 4772781 51 94100 87468 919817 6333536 10174e+ 06 11476523 145 4761512 51 94

Table 5 General parameters calculated for the design of pipe and well drainage

Parameter qc1 (mday) qc (mday) Qtot (m3day) β1β3 β2β4 βtβp ∆ε1 ∆ε2 ∆ε3Pipe drainage 2861lowast 10minus 3 3333lowast 10minus 3 16665 095 09 092Well drainage 2861lowast 10minus 3 3333lowast 10minus 3 16665 1 095 095 0911818 087667 minus001

Table 6 Pipe drainage optimal solution for a range of lifespan

Lifespan D (years) Pipe spacing E (m) Hydraulic head H (m) Total cost Z (RMB)1 2344577 13189 85823e+ 0610 2348422 13211 11711e+ 0750 2350565 13223 14698e+ 07100 2350644 13223 14840e+ 07


[2] C Xian Z Qi C S Tan and T-Q Zhang ldquoModeling hourlysubsurface drainage using steady-state and transientmethodsrdquo Journal of Hydrology vol 550 pp 516ndash526 2017

[3] M Akram A Azari A Nahvi Z Bakhtiari and H D SafaeeldquoSubsurface drainage in Khuzestan Iran environmentallyrevisited criteriardquo Irrigation and Drainage vol 62 no 3pp 306ndash314 2013

[4] S I Yannopoulos M E Grismer K M Bali andA N Angelakis ldquoEvolution of the materials and methodsused for subsurface drainage of agricultural lands from an-tiquity to the presentrdquo Water vol 12 no 6 p 1767 2020

[5] H S Acharya and D G Holsambre ldquoOptimum depth andspacing of subsurface drainsrdquo Journal of the Irrigation andDrainage Division vol 108 no 1 pp 77ndash80 1982

[6] A K Bhattacharya N Faroud S-T Chieng andR S Broughton ldquoSubsurface drainage cost and hydrologicmodelrdquo Journal of the Irrigation and Drainage Divisionvol 103 no 3 pp 299ndash308 1977

[7] J Boumans and L Smedema ldquoDerivation of cost-minimizingdepth for lateral pipe drainsrdquo Agricultural Water Manage-ment vol 12 no 1-2 pp 41ndash51 1986

[8] B R Chahar and G P Vadodaria ldquoOptimal spacing in anarray of fully penetrating ditches for subsurface drainagerdquoJournal of Irrigation and Drainage Engineering vol 136 no 1pp 63ndash67 2010

[9] D S Durnford T H Podmore and E V RichardsonldquoOptimal drain design for arid irrigated areasrdquo Transactionsof the ASAE vol 27 no 4 pp 1100ndash1105 1984

[10] H Ritzema H Nijland and F Croon ldquoSubsurface drainagepractices from manual installation to large-scale imple-mentationrdquo Agricultural Water Management vol 86 no 1-2pp 60ndash71 2006

[11] E HWiser R C Ward and D A Link ldquoOptimized design ofa subsurface drainage systemrdquo Transactions of the ASAEvol 17 no 1 pp 175ndash0178 1974

[12] L Cimorelli L Cozzolino C Covelli C MucherinoA Palumbo and D Pianese ldquoOptimal design of ruraldrainage networksrdquo Journal of Irrigation and Drainage En-gineering vol 139 no 2 pp 137ndash144 2013

[13] M Moradi-Jalal M A Marintildeo and A Afshar ldquoOptimaldesign and operation of irrigation pumping stationsrdquo Journalof Irrigation and Drainage Engineering vol 129 no 3pp 149ndash154 2003

[14] M Moradi-Jalal S I Rodin and M A Marintildeo ldquoUse ofgenetic algorithm in optimization of irrigation pumpingstationsrdquo Journal of Irrigation and Drainage Engineeringvol 130 no 5 pp 357ndash365 2004

[15] A K Sharma and P K Swamee ldquoCost considerations andgeneral principles in the optimal design of water distributionsystemsrdquo in Proceedings of the Water Distribution SystemsAnalysis Symposium Cincinnati OH USA August 2006

[16] M S Bennett and L W Mays ldquoOptimal design of detentionand drainage channel systemsrdquo Journal of Water ResourcesPlanning and Management vol 111 no 1 pp 99ndash112 1985

[17] B H Wall and A J Miller ldquoOptimization of parameters in amodel of soil water drainagerdquo Water Resources Researchvol 19 no 6 pp 1565ndash1572 1983

[18] A D Howard ldquoeoretical model of optimal drainage net-worksrdquo Water Resources Research vol 26 no 9pp 2107ndash2117 1990

[19] L Stuyt W Dierickx and J M Beltran Materials for Sub-surface Land Drainage Systems Food amp Agriculture Orga-nization Rome Italy 2005

[20] H Nijland F W Croon and H P Ritzema SubsurfaceDrainage Practices Guidelines for the Implementation Op-eration andMaintenance of Subsurface Pipe Drainage SystemsILRI Nairobi Kenya 2005

[21] R Broughton and J Fouss ldquoSubsurface drainage installationmachinery and methodsrdquo Agricultural Drainage vol 38pp 963ndash1003 1999

[22] G N Ganicheva ldquoSpecial works in municipal reclamationand water managementrdquo in Departmental Norms and Priceson Installation Repair and Construction WorksE M Bespalova Ed Preiskurantizdat Moscow Russia 1987


for the best surface drainage network design Amethodologywas presented by Chahar and Vadodaria [8] for deter-mining the optimal spacing of ditches fully penetrating intoan isotropic and homogeneous porous medium over animpervious layer Another methodology was introduced byMoradi-Jalal et al [13] for the optimal operation and designof pumping stations based on solving a nonlinear large-scaleprogramming problem And a new management model waspresented by Moradi-Jalal et al [14] for the optimal op-eration and design of water distribution systems An ob-jective function was constructed by Sharma and Swamee[15] for the cost structure of a pipe network system eoptimization model described by Bennett and Mays [16] wasbased upon dynamic programming for nonserial systemsthat determines the minimum cost of a drainage channelsystem and the detention for a watershed A nonlinearsquare fitting routine was used by Wall and Miller [17] todetermine the effective values of conductivity and moisturecharacteristics of the model of soil water drainage And asimulation model of the optimal drainage network waspresented by Howard [18] in which channels shift tominimize the total stream power within the network

is paper presents a novel strategy for the best design ofsubsurface horizontal and vertical drainage in an area ofdifferent saturated soil layers with different hydraulic con-ductivity and permeability and the permeability is in-creasing with depth With the use of modern optimizationalgorithms we can find the suitable values of groundwatertable drawdown and pipestubewells spacing that lead to theminimum cost of the total subsurface drainage system

2 Methodology

e least-cost design is that satisfying all design constraintswith the minimum total cost e objective function for pipeand well drainage was determined by considering all costcomponents that affect the drainage network design enconstraints were formulated depending on the hydraulicstudy of the study area e result was a nonlinear objectivefunction with nonlinear constraints A survey of modernoptimization algorisms was conducted to find the onesuitable for the solution of the formulated optimizationproblem It was found that the interior-point optimizationalgorithm was adapted to the problem and produced sat-isfactory results Two computer codes for both horizontaland vertical subsurface drainage were developed in MAT-LAB environment in order to derive the optimal solution forboth types of drainage systems e solutions then werecompered to know how the lifespan and type of the projectwill affect the total cost of the network design

3 Pipe Drainage Costs

e major cost components of pipe drainage system aredrainage materials installation and operation andmaintenance

31 DrainageMaterials e costs of drainage materials anddrainage pipe installation work and structures are major

components of the total cost of a pipe drain projectDrainage materials include collector and lateral pipes filterspump set and pump house outlet structures and manholesand outlet pitching

Collector and lateral pipes the collector pipes have atransmission function to carry water to the outlet under thegravitational flow so the UPVC (unplasticized poly vinylchloride) corrugated nonperforated pipe is used for thecollectors e pipe diameter of the collectors is chosendepending on the expected flow Single Wall PerforatedFlexible Corrugated UPVC Pipes (outer diameters of80ndash355mm) are widely used for lateral pipes

Artificial filtersenvelopes the filter or envelope materialaround the pipes plays an important role in preventing thefine particles of the soil from entering into the pipes with theflow Nowadays it is preferred to use artificial filters ratherthan gravel filters e artificial filters are cheaper in cost ofbetter quality and easier to handle and transport than gravelfilters For medium and light soil it is preferred to usenonwoven polypropylene fibers with a thickness of morethan 25mm and an opening size of more than 300 micronsand a needle punched geotextile nonwoven fabrics are usedfor heavy soil Perforated pipes precoated with industrialfilter are more preferred than locally coated pipes for qualityinstallation

Pump set and structures pump set and structure consistsof diesel pump set and pump house manholes (junctionbox) and outlet structures Other cross drain structures mayalso be required in larger projects but these are avoided forsimplicity e outlet structure and manholes are precastReinforced Cement Concrete (RCC) pipes of differentlengths and diameters depending on the site conditions Inthe case of pump outlets 900mm diameter manholes and1200mm outlet structures are used where 900mm diametermanholes are used in the gravity outlets e bottom edge ofthe RCC pipe for outlet structures and manholes is closed byfixed plate at the bottom Plastic coated iron bars are pro-vided in the walls of outlet structures and manholes to helpin inspection and cleaning In the case of gravity outletsinstead of RCC pipes plastic manholes and outlet pipes haverecently been used Plastic manholes and outlet pipes arerelatively expensive but they are easy to carry handle andinstall at the project sites [19] A small stone structure isconstructed at the outlet to protect the pipe end fromcollapsing under the conditions of gravity outlet A rodentguard is provided to prevent the pipes from getting stuck anddamaged by rodents that may enter the lateral pipes

32 Installation Installing the drainage system is thefunction of the installation unit according to the specifi-cations and design Supervisors must specify that the in-stallation (including drainage equipment) is carried outstrictly in accordance with specifications and design Var-ious drainage machines such as hydraulic excavatorstractor-mounted trenchers self-propelled trenchers withlaser automatic control and self-propelled machines with LorV plough and laser control are used to install collector andlateral pipes [20 21] Currently laser control self-propelled





zi Wi +1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857DlowastUi (1)





nm Atot

Elowast S (3)








ni Atot

Ai

(6)


Ai Elowast l (7)


Wi Atot


Atot


(8)

b

hcut

ϕ




Ui PWi + Uti (9)






Zi 1 +1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857DlowastP⎛⎝ ⎞⎠

Atot


+Atot


1 + E0( 1113857D

minus 1E0 1 + E0( 1113857

Dlowast

Atot

Elowast SlowastCs

(11)









Ui PWi + Uti + U

Ni (13)






UNi

9813600 η0




zi ni 1 +1 + E0( 1113857

Dminus 1


1 + E0( 1113857D

minus 1E0 1 + E0( 1113857


9813600 η0


5 Case Study










E 4

f2

+ TH

2qc

1113971

minus f⎛⎝ ⎞⎠ (17)

where

T 1113944 Ki hi( 1113857

f β1β2 hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

(18)


σ 0366 Logh

2πr sin((H + r)2h)



σt 0366 Loght

2πr sin (H + r)2ht( 1113857

(19)


N





H

2

ht hlowast1 + h2

(20)





3m

4m

12m

15m

2m

K1

K2

K3

K4

Ground waterinflow

Surfaceinflow

Surfaceoutflow

J1

J2

Ground wateroutflow

Rainfall



irrigationchannels

Evatranspiration



qc

h1

h2

h3

h4

hp

ht

E

Hm




Temperature 73 91 125 164 21 259 279 281 245 191 134 86 178



Lambert 12 17 24 26 57 79 105 95 65 38 14 11 48Ivanov 11 16 25 39 56 84 93 87 73 45 25 12 47



Wheat andbarley 1519 3528 5425 924 0 0 0 0 0 0 42 1519 2125



Q 1nlowastR

(23) lowastAlowastI

radic (21)




1geVge 015msec (22)



Km

(23)


qlowast



Minimize

Zi 1 +1 + E0( 1113857

Dminus 1

E 1 + E0( 1113857D

0





+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

Atot

Elowast SlowastCs

(25)

08

09

10

07

06

ε = 00250510

2030ε = 50ψ gt = 01

β

λ

0 02 04 06 08 09 094 098 10

(a)

08

09

10

ε = 503020

1005025

ψ = 001

ε = 0β

λ

0 02 04 06 08 09 094 098 10

(b)

08

09

10β

ψ lt = 0001 λε = 50

30201005025ε = 0

0 02 04 06 08 09 094 098 10

(c)



β1 β2 βt





Subject to

E 4

f2

+ TH

2qc

1113971



Km

H E gt 0

(26)

in which

f β1β2hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

σ 0366 Logh




σt 0366 Loght

2πr sin (2m + r)2ht( 1113857

(27)


hsw hc +qc1 B

2

Tρ + fc( 1113857 (28)

where

H hsw minus hc

H qc1 B

2

Tρ + fc( 1113857

ρ 0336 middot logB

πrc

fc β3β4k1

k3σc + βp

k2 minus k1

k3σcp +

k3 minus k2

k3σc3

(29)


σ 12π

1 minus x

xln07 lk

rc

+ ln1x

+Δε2

1113888 1113889 (30)

where

x lk

hi

(31)





qc1 p1 + p2

p1 φ + g

p2 104k4Δhh4

t

Δh p1

k1h1 minus m( 1113857

(32)



hchp

hth

h2

h3

B

hswLk

m2

H

qc2rc

m



(m + (Lk2))hi 01 015 02 025 03 04 05










h

(33)



Bor

G2

4q2c1

+ B2

11139741113972

minusG

2 qc1 (34)






n Qtot

Qi

(37)


nor L

Bor (38)


ni n minus nor (39)


tHc μπR

2H

2Qi

(40)






Minimize

Zi Qtot1 + 1 + E0( 1113857

Dminus 1E0 1 + E0( 1113857


Dminus 1E0 1 + E0( 1113857

D1113872 1113873lowastC2

qc1 lowastB2

⎡⎢⎣

+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

9813600 η0


(41)

Subject to

H qc1 lowast B

2

T0336 log

B

π rc

+ fc1113888 1113889

t μπ05652H

2 qc1

(42)








7 Conclusions



Data Availability




Acknowledgments


References



LifespanD (years)

Hydraulichead H (m)


(days)

Wellspacing B

(m)

Total cost Z(RMB)




Bor (m)



1 175848 1849211 8810597 29559e+ 05 22208976 75 7167423 34 4110 99645 1047862 6737206 70352e+ 05 12986063 128 5150194 47 8150 87810 923413 6345266 10038e+ 06 11519073 145 4772781 51 94100 87468 919817 6333536 10174e+ 06 11476523 145 4761512 51 94































zi Wi +1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857DlowastUi (1)





nm Atot

Elowast S (3)








ni Atot

Ai

(6)


Ai Elowast l (7)


Wi Atot


Atot


(8)

b

hcut

ϕ




Ui PWi + Uti (9)






Zi 1 +1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857DlowastP⎛⎝ ⎞⎠

Atot


+Atot


1 + E0( 1113857D

minus 1E0 1 + E0( 1113857

Dlowast

Atot

Elowast SlowastCs

(11)









Ui PWi + Uti + U

Ni (13)






UNi

9813600 η0




zi ni 1 +1 + E0( 1113857

Dminus 1


1 + E0( 1113857D

minus 1E0 1 + E0( 1113857


9813600 η0


5 Case Study










E 4

f2

+ TH

2qc

1113971

minus f⎛⎝ ⎞⎠ (17)

where

T 1113944 Ki hi( 1113857

f β1β2 hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

(18)


σ 0366 Logh

2πr sin((H + r)2h)



σt 0366 Loght

2πr sin (H + r)2ht( 1113857

(19)


N





H

2

ht hlowast1 + h2

(20)





3m

4m

12m

15m

2m

K1

K2

K3

K4

Ground waterinflow

Surfaceinflow

Surfaceoutflow

J1

J2

Ground wateroutflow

Rainfall



irrigationchannels

Evatranspiration



qc

h1

h2

h3

h4

hp

ht

E

Hm




Temperature 73 91 125 164 21 259 279 281 245 191 134 86 178



Lambert 12 17 24 26 57 79 105 95 65 38 14 11 48Ivanov 11 16 25 39 56 84 93 87 73 45 25 12 47



Wheat andbarley 1519 3528 5425 924 0 0 0 0 0 0 42 1519 2125



Q 1nlowastR

(23) lowastAlowastI

radic (21)




1geVge 015msec (22)



Km

(23)


qlowast



Minimize

Zi 1 +1 + E0( 1113857

Dminus 1

E 1 + E0( 1113857D

0





+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

Atot

Elowast SlowastCs

(25)

08

09

10

07

06

ε = 00250510

2030ε = 50ψ gt = 01

β

λ

0 02 04 06 08 09 094 098 10

(a)

08

09

10

ε = 503020

1005025

ψ = 001

ε = 0β

λ

0 02 04 06 08 09 094 098 10

(b)

08

09

10β

ψ lt = 0001 λε = 50

30201005025ε = 0

0 02 04 06 08 09 094 098 10

(c)



β1 β2 βt





Subject to

E 4

f2

+ TH

2qc

1113971



Km

H E gt 0

(26)

in which

f β1β2hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

σ 0366 Logh




σt 0366 Loght

2πr sin (2m + r)2ht( 1113857

(27)


hsw hc +qc1 B

2

Tρ + fc( 1113857 (28)

where

H hsw minus hc

H qc1 B

2

Tρ + fc( 1113857

ρ 0336 middot logB

πrc

fc β3β4k1

k3σc + βp

k2 minus k1

k3σcp +

k3 minus k2

k3σc3

(29)


σ 12π

1 minus x

xln07 lk

rc

+ ln1x

+Δε2

1113888 1113889 (30)

where

x lk

hi

(31)





qc1 p1 + p2

p1 φ + g

p2 104k4Δhh4

t

Δh p1

k1h1 minus m( 1113857

(32)



hchp

hth

h2

h3

B

hswLk

m2

H

qc2rc

m



(m + (Lk2))hi 01 015 02 025 03 04 05










h

(33)



Bor

G2

4q2c1

+ B2

11139741113972

minusG

2 qc1 (34)






n Qtot

Qi

(37)


nor L

Bor (38)


ni n minus nor (39)


tHc μπR

2H

2Qi

(40)






Minimize

Zi Qtot1 + 1 + E0( 1113857

Dminus 1E0 1 + E0( 1113857


Dminus 1E0 1 + E0( 1113857

D1113872 1113873lowastC2

qc1 lowastB2

⎡⎢⎣

+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

9813600 η0


(41)

Subject to

H qc1 lowast B

2

T0336 log

B

π rc

+ fc1113888 1113889

t μπ05652H

2 qc1

(42)








7 Conclusions



Data Availability




Acknowledgments


References



LifespanD (years)

Hydraulichead H (m)


(days)

Wellspacing B

(m)

Total cost Z(RMB)




Bor (m)



1 175848 1849211 8810597 29559e+ 05 22208976 75 7167423 34 4110 99645 1047862 6737206 70352e+ 05 12986063 128 5150194 47 8150 87810 923413 6345266 10038e+ 06 11519073 145 4772781 51 94100 87468 919817 6333536 10174e+ 06 11476523 145 4761512 51 94





























Ui PWi + Uti (9)






Zi 1 +1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857DlowastP⎛⎝ ⎞⎠

Atot


+Atot


1 + E0( 1113857D

minus 1E0 1 + E0( 1113857

Dlowast

Atot

Elowast SlowastCs

(11)









Ui PWi + Uti + U

Ni (13)






UNi

9813600 η0




zi ni 1 +1 + E0( 1113857

Dminus 1


1 + E0( 1113857D

minus 1E0 1 + E0( 1113857


9813600 η0


5 Case Study










E 4

f2

+ TH

2qc

1113971

minus f⎛⎝ ⎞⎠ (17)

where

T 1113944 Ki hi( 1113857

f β1β2 hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

(18)


σ 0366 Logh

2πr sin((H + r)2h)



σt 0366 Loght

2πr sin (H + r)2ht( 1113857

(19)


N





H

2

ht hlowast1 + h2

(20)





3m

4m

12m

15m

2m

K1

K2

K3

K4

Ground waterinflow

Surfaceinflow

Surfaceoutflow

J1

J2

Ground wateroutflow

Rainfall



irrigationchannels

Evatranspiration



qc

h1

h2

h3

h4

hp

ht

E

Hm




Temperature 73 91 125 164 21 259 279 281 245 191 134 86 178



Lambert 12 17 24 26 57 79 105 95 65 38 14 11 48Ivanov 11 16 25 39 56 84 93 87 73 45 25 12 47



Wheat andbarley 1519 3528 5425 924 0 0 0 0 0 0 42 1519 2125



Q 1nlowastR

(23) lowastAlowastI

radic (21)




1geVge 015msec (22)



Km

(23)


qlowast



Minimize

Zi 1 +1 + E0( 1113857

Dminus 1

E 1 + E0( 1113857D

0





+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

Atot

Elowast SlowastCs

(25)

08

09

10

07

06

ε = 00250510

2030ε = 50ψ gt = 01

β

λ

0 02 04 06 08 09 094 098 10

(a)

08

09

10

ε = 503020

1005025

ψ = 001

ε = 0β

λ

0 02 04 06 08 09 094 098 10

(b)

08

09

10β

ψ lt = 0001 λε = 50

30201005025ε = 0

0 02 04 06 08 09 094 098 10

(c)



β1 β2 βt





Subject to

E 4

f2

+ TH

2qc

1113971



Km

H E gt 0

(26)

in which

f β1β2hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

σ 0366 Logh




σt 0366 Loght

2πr sin (2m + r)2ht( 1113857

(27)


hsw hc +qc1 B

2

Tρ + fc( 1113857 (28)

where

H hsw minus hc

H qc1 B

2

Tρ + fc( 1113857

ρ 0336 middot logB

πrc

fc β3β4k1

k3σc + βp

k2 minus k1

k3σcp +

k3 minus k2

k3σc3

(29)


σ 12π

1 minus x

xln07 lk

rc

+ ln1x

+Δε2

1113888 1113889 (30)

where

x lk

hi

(31)





qc1 p1 + p2

p1 φ + g

p2 104k4Δhh4

t

Δh p1

k1h1 minus m( 1113857

(32)



hchp

hth

h2

h3

B

hswLk

m2

H

qc2rc

m



(m + (Lk2))hi 01 015 02 025 03 04 05










h

(33)



Bor

G2

4q2c1

+ B2

11139741113972

minusG

2 qc1 (34)






n Qtot

Qi

(37)


nor L

Bor (38)


ni n minus nor (39)


tHc μπR

2H

2Qi

(40)






Minimize

Zi Qtot1 + 1 + E0( 1113857

Dminus 1E0 1 + E0( 1113857


Dminus 1E0 1 + E0( 1113857

D1113872 1113873lowastC2

qc1 lowastB2

⎡⎢⎣

+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

9813600 η0


(41)

Subject to

H qc1 lowast B

2

T0336 log

B

π rc

+ fc1113888 1113889

t μπ05652H

2 qc1

(42)








7 Conclusions



Data Availability




Acknowledgments


References



LifespanD (years)

Hydraulichead H (m)


(days)

Wellspacing B

(m)

Total cost Z(RMB)




Bor (m)



1 175848 1849211 8810597 29559e+ 05 22208976 75 7167423 34 4110 99645 1047862 6737206 70352e+ 05 12986063 128 5150194 47 8150 87810 923413 6345266 10038e+ 06 11519073 145 4772781 51 94100 87468 919817 6333536 10174e+ 06 11476523 145 4761512 51 94


































E 4

f2

+ TH

2qc

1113971

minus f⎛⎝ ⎞⎠ (17)

where

T 1113944 Ki hi( 1113857

f β1β2 hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

(18)


σ 0366 Logh

2πr sin((H + r)2h)



σt 0366 Loght

2πr sin (H + r)2ht( 1113857

(19)


N





H

2

ht hlowast1 + h2

(20)





3m

4m

12m

15m

2m

K1

K2

K3

K4

Ground waterinflow

Surfaceinflow

Surfaceoutflow

J1

J2

Ground wateroutflow

Rainfall



irrigationchannels

Evatranspiration



qc

h1

h2

h3

h4

hp

ht

E

Hm




Temperature 73 91 125 164 21 259 279 281 245 191 134 86 178



Lambert 12 17 24 26 57 79 105 95 65 38 14 11 48Ivanov 11 16 25 39 56 84 93 87 73 45 25 12 47



Wheat andbarley 1519 3528 5425 924 0 0 0 0 0 0 42 1519 2125



Q 1nlowastR

(23) lowastAlowastI

radic (21)




1geVge 015msec (22)



Km

(23)


qlowast



Minimize

Zi 1 +1 + E0( 1113857

Dminus 1

E 1 + E0( 1113857D

0





+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

Atot

Elowast SlowastCs

(25)

08

09

10

07

06

ε = 00250510

2030ε = 50ψ gt = 01

β

λ

0 02 04 06 08 09 094 098 10

(a)

08

09

10

ε = 503020

1005025

ψ = 001

ε = 0β

λ

0 02 04 06 08 09 094 098 10

(b)

08

09

10β

ψ lt = 0001 λε = 50

30201005025ε = 0

0 02 04 06 08 09 094 098 10

(c)



β1 β2 βt





Subject to

E 4

f2

+ TH

2qc

1113971



Km

H E gt 0

(26)

in which

f β1β2hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

σ 0366 Logh




σt 0366 Loght

2πr sin (2m + r)2ht( 1113857

(27)


hsw hc +qc1 B

2

Tρ + fc( 1113857 (28)

where

H hsw minus hc

H qc1 B

2

Tρ + fc( 1113857

ρ 0336 middot logB

πrc

fc β3β4k1

k3σc + βp

k2 minus k1

k3σcp +

k3 minus k2

k3σc3

(29)


σ 12π

1 minus x

xln07 lk

rc

+ ln1x

+Δε2

1113888 1113889 (30)

where

x lk

hi

(31)





qc1 p1 + p2

p1 φ + g

p2 104k4Δhh4

t

Δh p1

k1h1 minus m( 1113857

(32)



hchp

hth

h2

h3

B

hswLk

m2

H

qc2rc

m



(m + (Lk2))hi 01 015 02 025 03 04 05










h

(33)



Bor

G2

4q2c1

+ B2

11139741113972

minusG

2 qc1 (34)






n Qtot

Qi

(37)


nor L

Bor (38)


ni n minus nor (39)


tHc μπR

2H

2Qi

(40)






Minimize

Zi Qtot1 + 1 + E0( 1113857

Dminus 1E0 1 + E0( 1113857


Dminus 1E0 1 + E0( 1113857

D1113872 1113873lowastC2

qc1 lowastB2

⎡⎢⎣

+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

9813600 η0


(41)

Subject to

H qc1 lowast B

2

T0336 log

B

π rc

+ fc1113888 1113889

t μπ05652H

2 qc1

(42)








7 Conclusions



Data Availability




Acknowledgments


References



LifespanD (years)

Hydraulichead H (m)


(days)

Wellspacing B

(m)

Total cost Z(RMB)




Bor (m)



1 175848 1849211 8810597 29559e+ 05 22208976 75 7167423 34 4110 99645 1047862 6737206 70352e+ 05 12986063 128 5150194 47 8150 87810 923413 6345266 10038e+ 06 11519073 145 4772781 51 94100 87468 919817 6333536 10174e+ 06 11476523 145 4761512 51 94






























H

2

ht hlowast1 + h2

(20)





3m

4m

12m

15m

2m

K1

K2

K3

K4

Ground waterinflow

Surfaceinflow

Surfaceoutflow

J1

J2

Ground wateroutflow

Rainfall



irrigationchannels

Evatranspiration



qc

h1

h2

h3

h4

hp

ht

E

Hm




Temperature 73 91 125 164 21 259 279 281 245 191 134 86 178



Lambert 12 17 24 26 57 79 105 95 65 38 14 11 48Ivanov 11 16 25 39 56 84 93 87 73 45 25 12 47



Wheat andbarley 1519 3528 5425 924 0 0 0 0 0 0 42 1519 2125



Q 1nlowastR

(23) lowastAlowastI

radic (21)




1geVge 015msec (22)



Km

(23)


qlowast



Minimize

Zi 1 +1 + E0( 1113857

Dminus 1

E 1 + E0( 1113857D

0





+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

Atot

Elowast SlowastCs

(25)

08

09

10

07

06

ε = 00250510

2030ε = 50ψ gt = 01

β

λ

0 02 04 06 08 09 094 098 10

(a)

08

09

10

ε = 503020

1005025

ψ = 001

ε = 0β

λ

0 02 04 06 08 09 094 098 10

(b)

08

09

10β

ψ lt = 0001 λε = 50

30201005025ε = 0

0 02 04 06 08 09 094 098 10

(c)



β1 β2 βt





Subject to

E 4

f2

+ TH

2qc

1113971



Km

H E gt 0

(26)

in which

f β1β2hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

σ 0366 Logh




σt 0366 Loght

2πr sin (2m + r)2ht( 1113857

(27)


hsw hc +qc1 B

2

Tρ + fc( 1113857 (28)

where

H hsw minus hc

H qc1 B

2

Tρ + fc( 1113857

ρ 0336 middot logB

πrc

fc β3β4k1

k3σc + βp

k2 minus k1

k3σcp +

k3 minus k2

k3σc3

(29)


σ 12π

1 minus x

xln07 lk

rc

+ ln1x

+Δε2

1113888 1113889 (30)

where

x lk

hi

(31)





qc1 p1 + p2

p1 φ + g

p2 104k4Δhh4

t

Δh p1

k1h1 minus m( 1113857

(32)



hchp

hth

h2

h3

B

hswLk

m2

H

qc2rc

m



(m + (Lk2))hi 01 015 02 025 03 04 05










h

(33)



Bor

G2

4q2c1

+ B2

11139741113972

minusG

2 qc1 (34)






n Qtot

Qi

(37)


nor L

Bor (38)


ni n minus nor (39)


tHc μπR

2H

2Qi

(40)






Minimize

Zi Qtot1 + 1 + E0( 1113857

Dminus 1E0 1 + E0( 1113857


Dminus 1E0 1 + E0( 1113857

D1113872 1113873lowastC2

qc1 lowastB2

⎡⎢⎣

+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

9813600 η0


(41)

Subject to

H qc1 lowast B

2

T0336 log

B

π rc

+ fc1113888 1113889

t μπ05652H

2 qc1

(42)








7 Conclusions



Data Availability




Acknowledgments


References



LifespanD (years)

Hydraulichead H (m)


(days)

Wellspacing B

(m)

Total cost Z(RMB)




Bor (m)



1 175848 1849211 8810597 29559e+ 05 22208976 75 7167423 34 4110 99645 1047862 6737206 70352e+ 05 12986063 128 5150194 47 8150 87810 923413 6345266 10038e+ 06 11519073 145 4772781 51 94100 87468 919817 6333536 10174e+ 06 11476523 145 4761512 51 94




























Q 1nlowastR

(23) lowastAlowastI

radic (21)




1geVge 015msec (22)



Km

(23)


qlowast



Minimize

Zi 1 +1 + E0( 1113857

Dminus 1

E 1 + E0( 1113857D

0





+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

Atot

Elowast SlowastCs

(25)

08

09

10

07

06

ε = 00250510

2030ε = 50ψ gt = 01

β

λ

0 02 04 06 08 09 094 098 10

(a)

08

09

10

ε = 503020

1005025

ψ = 001

ε = 0β

λ

0 02 04 06 08 09 094 098 10

(b)

08

09

10β

ψ lt = 0001 λε = 50

30201005025ε = 0

0 02 04 06 08 09 094 098 10

(c)



β1 β2 βt





Subject to

E 4

f2

+ TH

2qc

1113971



Km

H E gt 0

(26)

in which

f β1β2hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

σ 0366 Logh




σt 0366 Loght

2πr sin (2m + r)2ht( 1113857

(27)


hsw hc +qc1 B

2

Tρ + fc( 1113857 (28)

where

H hsw minus hc

H qc1 B

2

Tρ + fc( 1113857

ρ 0336 middot logB

πrc

fc β3β4k1

k3σc + βp

k2 minus k1

k3σcp +

k3 minus k2

k3σc3

(29)


σ 12π

1 minus x

xln07 lk

rc

+ ln1x

+Δε2

1113888 1113889 (30)

where

x lk

hi

(31)





qc1 p1 + p2

p1 φ + g

p2 104k4Δhh4

t

Δh p1

k1h1 minus m( 1113857

(32)



hchp

hth

h2

h3

B

hswLk

m2

H

qc2rc

m



(m + (Lk2))hi 01 015 02 025 03 04 05










h

(33)



Bor

G2

4q2c1

+ B2

11139741113972

minusG

2 qc1 (34)






n Qtot

Qi

(37)


nor L

Bor (38)


ni n minus nor (39)


tHc μπR

2H

2Qi

(40)






Minimize

Zi Qtot1 + 1 + E0( 1113857

Dminus 1E0 1 + E0( 1113857


Dminus 1E0 1 + E0( 1113857

D1113872 1113873lowastC2

qc1 lowastB2

⎡⎢⎣

+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

9813600 η0


(41)

Subject to

H qc1 lowast B

2

T0336 log

B

π rc

+ fc1113888 1113889

t μπ05652H

2 qc1

(42)








7 Conclusions



Data Availability




Acknowledgments


References



LifespanD (years)

Hydraulichead H (m)


(days)

Wellspacing B

(m)

Total cost Z(RMB)




Bor (m)



1 175848 1849211 8810597 29559e+ 05 22208976 75 7167423 34 4110 99645 1047862 6737206 70352e+ 05 12986063 128 5150194 47 8150 87810 923413 6345266 10038e+ 06 11519073 145 4772781 51 94100 87468 919817 6333536 10174e+ 06 11476523 145 4761512 51 94




























Subject to

E 4

f2

+ TH

2qc

1113971



Km

H E gt 0

(26)

in which

f β1β2hK3

K11113888 1113889σ + βtht

K2 minus K3( 1113857

K11113896 1113897σt +

K1 minus K2( 1113857

K11113896 1113897σ1h

lowast1

σ 0366 Logh




σt 0366 Loght

2πr sin (2m + r)2ht( 1113857

(27)


hsw hc +qc1 B

2

Tρ + fc( 1113857 (28)

where

H hsw minus hc

H qc1 B

2

Tρ + fc( 1113857

ρ 0336 middot logB

πrc

fc β3β4k1

k3σc + βp

k2 minus k1

k3σcp +

k3 minus k2

k3σc3

(29)


σ 12π

1 minus x

xln07 lk

rc

+ ln1x

+Δε2

1113888 1113889 (30)

where

x lk

hi

(31)





qc1 p1 + p2

p1 φ + g

p2 104k4Δhh4

t

Δh p1

k1h1 minus m( 1113857

(32)



hchp

hth

h2

h3

B

hswLk

m2

H

qc2rc

m



(m + (Lk2))hi 01 015 02 025 03 04 05










h

(33)



Bor

G2

4q2c1

+ B2

11139741113972

minusG

2 qc1 (34)






n Qtot

Qi

(37)


nor L

Bor (38)


ni n minus nor (39)


tHc μπR

2H

2Qi

(40)






Minimize

Zi Qtot1 + 1 + E0( 1113857

Dminus 1E0 1 + E0( 1113857


Dminus 1E0 1 + E0( 1113857

D1113872 1113873lowastC2

qc1 lowastB2

⎡⎢⎣

+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

9813600 η0


(41)

Subject to

H qc1 lowast B

2

T0336 log

B

π rc

+ fc1113888 1113889

t μπ05652H

2 qc1

(42)








7 Conclusions



Data Availability




Acknowledgments


References



LifespanD (years)

Hydraulichead H (m)


(days)

Wellspacing B

(m)

Total cost Z(RMB)




Bor (m)



1 175848 1849211 8810597 29559e+ 05 22208976 75 7167423 34 4110 99645 1047862 6737206 70352e+ 05 12986063 128 5150194 47 8150 87810 923413 6345266 10038e+ 06 11519073 145 4772781 51 94100 87468 919817 6333536 10174e+ 06 11476523 145 4761512 51 94































h

(33)



Bor

G2

4q2c1

+ B2

11139741113972

minusG

2 qc1 (34)






n Qtot

Qi

(37)


nor L

Bor (38)


ni n minus nor (39)


tHc μπR

2H

2Qi

(40)






Minimize

Zi Qtot1 + 1 + E0( 1113857

Dminus 1E0 1 + E0( 1113857


Dminus 1E0 1 + E0( 1113857

D1113872 1113873lowastC2

qc1 lowastB2

⎡⎢⎣

+1 + E0( 1113857

Dminus 1

E0 1 + E0( 1113857Dlowast

9813600 η0


(41)

Subject to

H qc1 lowast B

2

T0336 log

B

π rc

+ fc1113888 1113889

t μπ05652H

2 qc1

(42)








7 Conclusions



Data Availability




Acknowledgments


References



LifespanD (years)

Hydraulichead H (m)


(days)

Wellspacing B

(m)

Total cost Z(RMB)




Bor (m)



1 175848 1849211 8810597 29559e+ 05 22208976 75 7167423 34 4110 99645 1047862 6737206 70352e+ 05 12986063 128 5150194 47 8150 87810 923413 6345266 10038e+ 06 11519073 145 4772781 51 94100 87468 919817 6333536 10174e+ 06 11476523 145 4761512 51 94





























7 Conclusions



Data Availability




Acknowledgments


References



LifespanD (years)

Hydraulichead H (m)


(days)

Wellspacing B

(m)

Total cost Z(RMB)




Bor (m)



1 175848 1849211 8810597 29559e+ 05 22208976 75 7167423 34 4110 99645 1047862 6737206 70352e+ 05 12986063 128 5150194 47 8150 87810 923413 6345266 10038e+ 06 11519073 145 4772781 51 94100 87468 919817 6333536 10174e+ 06 11476523 145 4761512 51 94


















































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