Article Request Page ASABE Journal Article ## Choice of Pipe Material Influences Drain Spacing and System Cost in Subsurface Drainage Design

Ehsan Ghane^{1,*}## Published in

Applied Engineering in Agriculture38(4): 685-695 (doi: 10.13031/aea.15053). Copyright 2022 American Society of Agricultural and Biological Engineers.The authors have paid for open access for this article. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License https://creative commons.org/licenses/by-nc-nd/4.0.

Submitted for review on 9 February 2022 as manuscript number NRES 15053; approved for publication as a Research Brief by Associate Editor Dr. Michael Dukes and Community Editor Dr. Kati Migliaccio of the Natural Resources & Environmental Systems Community of ASABE on 7 June 2022.

^{1 }Biosystems and Agricultural Engineering, Michigan State University, East Lansing, Michigan, USA.

^{* }Correspondence: ghane@msu.edu

Highlights

For 3- or 4-row regular-perforated pipes, the effective radius ranged from 0.3 to 0.9 cm (average 0.6 cm).For 8-row regular-perforated pipes, the effective radius was 1.9 cm.The effective radius of sock-wrapped pipes ranged from 5.7 to 6.0 cm (average 5.9 cm).The 8-row sand-slot pipes had a lower initial system cost than the sock-wrapped pipes.The 8-row regular-perforated pipes had a lower initial system cost than the 4-row regular-perforated pipes.

Abstract.Knitted-sock envelopes are applied in agricultural subsurface drainage to prevent sediment clogging of the drain pipes. In the United States and Canada, sand-slot pipes are sometimes used as a cheaper alternative to sock-wrapped pipes. However, their initial system cost has not been compared. The main objective of this study was to evaluate the effect of pipe material on drain spacing and initial system cost. First, the theoretical effective radius of each pipe material was estimated. Then, the drain spacing was calculated for each pipe material in a drainage design, such that each pipe would provide the same design drainage rate (i.e., same water removal rate). The results showed that the effective radius of sock-wrapped pipes (average 5.9 cm) was much greater than that of 4-row (average 0.4 cm) and 8-row perforated sand-slot pipes (average 1.6 cm). Rows refer to number of longitudinal rows of perforations. The sock-wrapped pipes considerably increased the effective radius of the pipe by reducing the entrance head loss. Furthermore, the sock-wrapped pipes allowed for a wider drain spacing (ranging from 0.8 to 5.4 m wider) in soil with risk of drain sedimentation, thereby reducing the total length of the lateral drain pipe needed for drainage design compared to both 4- and 8-row sand-slot pipes. The 8-row regular-perforated pipes allowed for a wider optimum drain spacing, thereby reducing the initial system cost in soil without a drain sedimentation issue compared to 4-row regular-perforated pipes. In conclusion, even though the sock-wrapped pipe reduced the total length of the lateral drain pipe, the 8-row sand-slot pipe had a lower initial system cost compared to the sock-wrapped pipe, when designed at the same design drainage rate and drain depth.

Drain spacing, Effective radius, Entrance resistance, Geotextile, Knitted sock, Perforation. Keywords.In temperate, humid regions, subsurface drainage is used to remove excess water from poorly drained soils, thereby providing suitable conditions for crop production (Fausey et al., 1995; Evans and Fausey, 1999). In these systems, water from the soil profile enters perforated, lateral drain pipes and is conveyed by a network of submains and mains to an outlet. Drain spacing is one of the important design parameters that affects water removal and water quality in drainage systems (Skaggs, 2017; Ghane and Askar, 2021). In humid regions, this important design parameter is commonly determined using the Hooghoudt (1940) steady-state equation (Ayars and Evans, 2015).

The Hooghoudt (1940) equation was originally developed for an ideal drain pipe, which has a fully permeable wall (Skaggs, 1978). To account for radial head loss due to convergence of flowlines towards the drain, Hooghoudt replaced the distance between the drain and restrictive layer with an “equivalent depth”,

d(Skaggs, 1991). Additionally, real drain pipes have a finite number of perforations that cause entrance head loss due to convergence of flowlines toward the drain perforations (Wesseling and Homma, 1967; Skaggs, 1978; Dierickx and van der Molen, 1981). To account for entrance head loss, Childs and Youngs (1958) substituted the real drain radius with a smaller drain radius described as “effective radius”. The effective radius is the radius of an imaginary pipe with a completely open wall, so that an imaginary open-wall pipe with effective radius_{e }Rhas the same entrance resistance as the real drain (Skaggs, 1978). In drainage design, the effective radius of a pipe is used to calculate the equivalent depth (Moody, 1966; van der Molen and Wesseling, 1991), which is then used in the Hooghoudt equation to determine the drain spacing._{ef}The effective radius of a drain pipe can be calculated based on the entrance resistance factor (Dierickx, 1999). This factor is a constant that depends on the drain perforation characteristics (i.e., shape and pattern) and the drain envelope material (Dierickx, 1999; Stuyt and Dierickx, 2006). It has long been known that the use of a drain envelope reduces the entrance resistance. In an early study, Dennis and Trafford (1975) found that placement of a thin gravel envelope around a clay tile doubled the drain spacing by reducing the entrance resistance and allowing increased flow into the drain pipe.

There are three types of drain envelopes: granular mineral (e.g., gravel, coarse sand, crushed stone), organic (e.g., rice straw, woodchips, cereal straw, coconut fiber), and synthetic (Stuyt and Dierickx, 2006). Synthetic envelopes come in two types: pre-wrapped loose materials and pre-wrapped geotextiles (Stuyt et al., 2005). Pre-wrapped geotextile envelopes come in three types, woven, non-woven, and knitted, with the knitted being the most common in the U.S. and Canada (Stuyt et al., 2005). Geotextiles have two key properties that make them suitable as an envelope material: they allow for faster water entry into the drain pipes, and they prevent sediment clogging of the drain pipes (Rollin et al., 1987; Stuyt et al., 2005). Although some geotextile envelopes are thin, they substantially decrease the entrance resistance, thereby increasing the effective radius (Oyarce et al., 2017; Stuyt and Dierickx, 2006).

Knitted-sock geotextile is the most common type of geotextile envelopes used for agricultural subsurface drainage in the U.S. and Canada. This type of envelope has also been used to a lesser extent since the 1980s in the United Kingdom, Netherlands, France, Belgium, Germany, Sweden, and Denmark, and more recently in Poland and the Baltic nations (Paul Mutter, 2021, CARRIFF Canada, personal communication). However, to our knowledge, no study has quantified the effective radius of a pipe that is wrapped with knitted-sock geotextile. The effective radius is needed to calculate the drain spacing accurately.

The drainage industry in the U.S. and Canada has manufactured sand-slot pipes (i.e., narrow-slot or knife-cut pipes) as a cheaper alternative to sock-wrapped pipes to prevent sediment clogging of the drain pipes. However, their drain spacing and initial system cost has not been compared. This information is important as it informs the practitioner as to which pipe allows for a wider drain spacing and requires a lower initial system cost. Therefore, there is a need to compare the drain spacing and initial system cost between sand-slot and sock-wrapped pipes.

The objectives of this study were: (1) to quantify the effective radius of commercially available pipes with and without a knitted-sock envelope, (2) to compare the drain spacing and initial system cost of sock-wrapped pipes versus sand-slot pipes that are used in soil with risk of drain sedimentation, and (3) to compare the drain spacing and initial system cost of 8-row regular-perforated pipes versus 4-row regular-perforated pipes that are used in soil with no risk of drain sedimentation. The value of this study is that it provides effective radii of drain pipes required for drainage design and modeling, and informs drainage design decisions as to which pipe material is more suitable for site-specific conditions.

Materials and Methods

Characterizing the Perforation Shape and Pattern of PipesThe first step in determining the effective radius of sock-wrapped pipes is to characterize the perforation shape and pattern of corrugated perforated pipes without a knitted sock. In 2019, we obtained 3-m-long samples of 3-, 4-, and 8-row 100-mm diameter agricultural subsurface drain pipes (referred to as “regular-perforated”) from seven manufacturers in the United States. Rows refer to number of longitudinal rows of perforations. Regular-perforated pipes are used when there is no risk of drain sedimentation. We obtained a sample of a 4-row regular-perforated 150-mm diameter pipe from Timewell Drainage Products (Timewell, Ill.). We also obtained 3-m long samples of 4-, 6-, and 8-row 100-mm diameter sand-slot pipes (i.e., narrow-slot or knife-cut) from the same manufacturers. The sand-slot pipes have a narrow perforation width to prevent sediment clogging of the drain pipes. The regular-perforated and sand-slot pipes are manufactured following industry standards (ASTM F-405, ASTM F-667, and AASHTO M-252).

A digital caliper (Mitutoyo 500-753-20, Japan) was used to measure the perforation dimensions shown in figure 1. The perforation length (?

) was measured based on a standard method for measuring pipe dimension (AASHTO M 252-09, 2012). Each of the measurements were repeated 8 to 24 times along the length of each drain pipe, and then the average value was used to represent each dimension. Perforation area was calculated as the multiplication of average length and width. Drain pipe photos and detailed procedures for characterizing the perforation dimensions are presented in the Supplementary Sections S1 and S2._{p}

Calculating the Entrance-Resistance Factor of PipesThe perforation characteristics of each pipe were used to calculate the theoretical entrance-resistance factor. Dierickx (1980) analytically modeled the entrance resistance of perforated pipes and derived analytical solutions to calculate the dimensionless entrance-resistance factor (a

) for various corrugation and perforation types. These analytical solutions have been shown to provide sufficiently accurate entrance-resistance values in experimental studies (Dierickx, 1980, 1999; Oyarce et al., 2017). The analytical equation for corrugated pipes with perforations in the valleys (fig. 1) is written as:_{e}(1)

where

c= perforation spacing (cm),

R= outer radius of the pipe (cm),_{o}

R= inner radius of the pipe (cm),_{i}?

= valley width (cm),_{v}?

= perforation width (cm),_{s}?

= perforation length (cm), and_{p}

N= number of longitudinal rows of perforation (fig. 1).

Figure 1. (left) Diagram of flowlines and distances required to characterize the perforations [Reprinted from Dierickx (1999) with permission from Wiley]. ßis the perforation width, ?_{s}is the perforation length,_{p}ßis the valley width,_{v }cis the perforation spacing,Ris the inner radius of the drain, and_{i}Ris the outer radius of the pipe. (right) Diagram of an 8-row pipe and cross-section view of the pipe illustrating the number of longitudinal rows of perforations (_{o}N).

Calculating the Effective Radius of PipesThe entrance-resistance factor was determined from equation 1 to calculate the theoretical effective radius of pipes with the analytical equation developed by Dierickx (1999). This equation is:

(2)

where

R= effective radius of the pipe (cm),_{ef}

R= outer radius of the pipe (cm) (fig. 1), and_{o}a

= the entrance-resistance factor (dimensionless)._{e}

Calculating the Effective Radius of Sock-wrapped PipesDierickx (1999) derived an analytical equation to calculate the effective radius of sock-wrapped pipes. This equation is:

(3)

where

d= thickness of the knitted-sock envelope (cm),_{ks}

K= saturated hydraulic conductivity of the soil_{sd}surrounding the drain pipe (cm/h), and

K= saturated hydraulic conductivity of the knitted-sock_{ks}envelope (cm/h).

As of 2021, the pipe manufacturers of ADS (Hilliard, Ohio), Baughman (Paulding, Ohio), Fratco (Monticello, Ind.), Prinsco (Willmar, Minn.), Springfield (Auburn, Ill.), and Timewell (Timewell, Ill.) used type A circular-knitted socks provided by CARRIFF (London, Ont., Canada). See Supplementary Section S3 for a photo of the CARRIFF sock circular-knitted openings. Haviland used the S4 black knitted sock provided by SYFILCO Ltd. (Exeter, Ont., Canada) (table 1).

Determining the Drain Spacing and Initial System Cost of Pipe MaterialsThe effects of pipe material on drain spacing and initial system cost were evaluated for each combination of three factors: soil lateral saturated hydraulic conductivity, drain depth (75 and 125 cm), and depth to restrictive layer (180 and 350 cm). The pipe material comparison was sock-wrapped versus 4-row sand-slot, sock-wrapped versus 8-row sand-slot, and 8-row regular-perforated versus 4-row regular-perforated pipes. When comparing sock-wrapped versus sand-slot pipes, sandy loam and silt loam soils were considered, which can cause sediment clogging of the drain pipes and have lateral saturated hydraulic conductivities (

K) ranging from approximately 2.0 to 6.0 cm/h (Vlotman et al., 2020). When comparing 8-row regular-perforated versus 4-row regular-perforated pipes, clay loam and silty clay loam soils were considered, which generally do not cause sediment clogging of the drain pipes and have lateral saturated hydraulic conductivities ranging from approximately 0.4 to 1.7 cm/h (Vlotman et al., 2020). The initial system cost for each combination of factors was calculated in the following steps (fig. 2)._{e}

Table 1. Summary of polyester circular-knitted sock specifications provided by the manufacturers. Property Test Method CARRIFF

Type A Knitted SockSYFILCO S4

Black Knitted SockMass per unit area (g/m ^{2}) (average)ASTM D3776 105 101 Filtration opening size, FOS, O(mm) (max)_{95 }^{[a]}CAN/CGSB-148.1 0.45 0.45 Apparent opening size, AOS (mm) (max) ASTM D4751 0.60 0.60 Thickness, d(mm) (min)_{ks}ASTM D4491 0.75 0.75 Saturated hydraulic conductivity, K(cm/h) (min)_{ks}ASTM D4491 1404 1440

^{[a]}O= envelope pore size at which 95% of pores are smaller, which is analogous to filtration openings size (Vlotman et al., 2000)._{95}

Figure 2. The diagram of the sequence of steps taken to determine the initial system cost of various pipe materials.

Step 1:The design drainage rate (DDR) is the depth of water that the drainage system is designed to remove from the soil in 24 h in units of cm per day. TheDDRfor each combination of factors was calculated using the empirical equation developed by Ghane et al. (2021) that maximizes economic return on investment based on site-specific conditions. The empirical equation was developed based on DRAINMOD simulations for a wide range of conditions including three drain depths and five soil types at four locations in the Midwest United States. This empirical equation is:(4)

where

DDR= design drainage rate defined as the rate when water

table midway between lateral drain pipes is at the

soil surface (cm/day),

P= long-term average growing-season precipitation_{g}from one month prior to planting to four months

after planting (mm),

D= drain depth from the soil surface to the drain center_{d}(cm)

K= equivalent saturated hydraulic conductivity of the_{e}soil profile (cm/day), and

D= depth to restrictive layer from the soil surface (cm).Equation 4 estimates the

DDRthat maximizes economic return on investment based on the known variablesP,_{g}D,_{d}K, and_{e}D. To proceed with calculatingDDR, aPof 438 mm was assumed. The findings of this article are valid for other assumptions of_{g}P. This assumption is merely a means to calculate the_{g}DDRfor Step 2. As an alternative to Step 1, one can assume a constant value forDDRand the results of this paper will still be the same.

Step 2:With theDDRfrom Step 1, the optimum drain spacing was estimated using the Hooghoudt equation (Hooghoudt, 1940). When the water table midway between subsurface drain pipes is at the soil surface, the steady-state drainage rate is equal to theDDR(Skaggs, 2007; Skaggs et al., 2006). Based on this definition, the Hooghoudt equation can be written as:(5)

where

S= drain spacing (cm),

DDR= design drainage rate (cm/day),

d= equivalent depth (cm),_{e}

D= drain depth from the soil surface to the_{d}drain center (cm), and

K= equivalent saturated hydraulic conductivity of the_{e}soil profile (cm/day) (fig. 3).

The equivalent depth was calculated from the analytical solution developed by van der Molen and Wesseling (1991). This equation is:

(6)

when

x =2??d/Sand x < 0.5,F(x) can be approximated by:(7)

where

d= equivalent depth (cm), and_{e}

d= distance from the drain center to the restrictivelayer (cm).

Figure 3. Diagram of the soil and water-table profile with subsurface drainage. Ris the rainfall rate that is equal to the design drainage rate (DDR) when the midway water table is at the soil surface,dis the equivalent depth,_{e}Dis the drain depth,_{d}Sis the drain spacing,Dis the depth to the restrictive layer, anddis the distance from the drain center to the restrictive layer.

Step 3:The total length of 100-mm diameter lateral drain pipe required to install the drainage system in a field with a unit area of 1 ha was calculated. The length was calculated per unit hectare by dividing 10000 m^{2}by the optimum drain spacing in meters.

Step 4:The initial cost of the drainage system was calculated as the sum of the pipe material cost and installation cost. The cost of 100-mm diameter pipe material was $1.02/m. The cost of 100-mm diameter pipe material with CARRIFF sock was $1.48/m. The cost of pipe installation (with or without sock) was $1.64/m. These costs were determined from Michigan drainage contractors and drain pipe manufacturers in 2020. To calculate the initial system cost/ha, the sum of material and installation cost for pipe without sock ($2.66/m) or with sock ($3.12/m) was multiplied by the total length of 100-mm diameter lateral drain pipe required per hectare.

Results and Discussion

The Effective Radius of 3- and 4-row Regular-perforated PipesThe 3- and 4-row regular-perforated pipes are installed in soil with high clay and organic matter (cohesive soil), which generally does not cause sediment clogging of the drain pipes. The entrance-resistance factors and the effective radii of the pipes were calculated based on the measured perforation dimensions (tables 2 and 3). The effective radius varied from 0.3 to 0.9 cm (average 0.6 cm) for a wide range of perforation dimensions. For drainage design and modeling, an average effective radius of about 0.6 cm can be used for 3- or 4-row regular-perforated pipes.

Even though effective radius of 3- and 4-row pipes varied among manufacturers (ranging from 0.3 to 0.9 cm), the difference in their effective radius was small. Such a small difference in effective radius had a negligible effect on drain spacing. To demonstrate this with an example, see Supplementary Section S4.

It is important to note that the calculated effective radii in table 2 are based on 3-m-long samples obtained from the manufacturer in 2019. During the drain pipe manufacturing, minor variation in perforation slot width and length may occur. That is why pipe manufacturers maintain the perforation characteristics within an acceptable range of variation. These minor variations in perforation characteristics do not have a considerable effect on effective radius, thereby having a negligible effect on drain spacing and cost. Effective radius is affected considerably when the manufacturer re-designs their perforation characteristics to upgrade from a 4-row to an 8-row perforated pipe. In a subsequent section, the number of rows of perforation has been identified as the most important perforation characteristic that considerably affects effective radius.

The Effective Radius of 8-row Regular-perforated PipesThe effective radius of the 8-row regular-perforated pipes (No. 4R and 8R) was 1.9 cm, which is considerably larger than that of the 3- and 4-row pipes (table 2). Mohammad and Skaggs (1983) experimentally measured a much larger effective radius of 3.9 cm for a 100-mm diameter 8-row regular-perforated corrugated pipe with a perforation area of 182 cm

^{2}/m based on radial flow and heat transfer theory. The difference in effective radius between their study and our study can be explained by the differences in perforation characteristics (i.e., shape and pattern) between the drain pipes used by Mohammad and Skaggs (1983) and those used in our study. For drainage design and modeling, an effective radius of 1.9 cm can be used for 8-row regular-perforated pipes.

Table 2. Summary of key perforation characteristics, entrance-resistance factors, and effective radius of 100-mm diameter corrugated regular-perforated pipes manufactured in the USA. All measurements are average values. ^{[a]}Pipe Make Pipe

No.Pipe Type NNo. Perf.

per Meter?? _{s}

(cm)? _{p}

(cm)?? _{v}

(cm)c

(cm)R_{o}

(cm)R_{i}

(cm)Perf.

Area

(cm^{2}/m)Entrance

Resistance

Factor, a_{e}Effective

Radius,R_{ef}

(cm)ADS 1R Single-Wall Corrugated HDPE Regular Perf 3 174 0.16 2.08 0.81 1.75 5.86 5.16 58 0.33 0.7 Baughman

Tile2R Single-Wall 3/32

Staggered (Regular Perf)4 248 0.21 1.36 0.61 1.62 5.90 5.04 71 0.46 0.3 3R Single-Wall 1/16

Slot (Regular Perf)4 248 0.16 1.95 0.61 1.67 5.90 5.04 77 0.35 0.7 4R Single-Wall 8-Slot 8 488 0.23 1.57 0.64 1.62 5.90 5.04 176 0.18 1.9 Fratco 5R SingleCorr Single-Wall

Slot (Regular Perf)4 240 0.16 1.97 0.65 1.72 5.89 5.14 76 0.31 0.9 Haviland

Drainage6R 416 Staggered

(Regular Perf)4 236 0.20 1.82 0.55 1.70 5.89 5.11 86 0.36 0.6 7R 418 Staggered

(Regular Perf)4 240 0.33 1.91 0.53 1.64 5.89 5.11 151 0.32 0.8 8R 416 8-Row 8 480 0.19 1.82 0.55 1.70 5.89 5.11 166 0.18 1.9 Prinsco 9R Goldline Single-Wall

Regular Perforation4 232 0.14 1.92 0.59 1.73 5.90 5.13 62 0.35 0.6 Springfield

Plastics10R Single-Wall Regular

Slotted (Regular Perf)4 240 0.12 1.97 0.50 1.65 5.82 5.01 57 0.39 0.5 Timewell

Drainage11R Single-Wall HDPE

Regular Perforated4 228 0.15 1.87 0.52 1.76 5.80 5.01 64 0.41 0.4

^{[a]}??is the perforation width, ?_{s }is the perforation length, ??_{p}is the valley width,_{v }cis the perforation spacing,Nis the number longitudinal rows of

perforations,Ris the outer radius of the pipe, and_{o }Ris the inner radius of the pipe (see fig. 1)._{i}

Table 3. Summary of key perforation characteristics, entrance-resistance factors, and effective radius of 100-mm diameter corrugated sand-slot pipes manufactured in the USA. All measurements are average values. Pipe Make Pipe

No.Pipe Type NNo. Perf.

per Meter?? _{s}

(cm)? _{p}

(cm)?? _{v}

(cm)c

(cm)R_{o}

(cm)R_{i}

(cm)Perf.

Area

(cm^{2}/m)Entrance

Resistance

Factor, a_{e}Effective

Radius,R_{ef}

(cm)ADS 1S Single-Wall

HDPE Sand Slot4 232 0.05 1.95 0.79 1.74 5.86 5.16 23 0.35 0.7 Baughman Tile 2S Single-Wall Knife

Cut (Sand Slot)6 360 0.07 1.94 0.67 1.69 5.90 5.04 49 0.25 1.2 Fratco 3S SingleCorr Single-Wall

Narrow Slot (Sand Slot)4 240 0.06 1.88 0.63 1.68 5.89 5.14 27 0.39 0.5 Haviland Drainage 4S Corrugated

Sand Slot8 496 0.05 1.95 0.53 1.65 5.89 5.11 48 0.21 1.6 Prinsco 5S Goldline Single-Wall

Narrow Slot (Sand Slot)4 232 0.06 1.92 0.56 1.75 5.90 5.13 27 0.43 0.4 Springfield Plastics 6S Single-Wall Ultra-

Narrow (Sand Slot)4 240 0.05 1.38 0.54 1.66 5.82 5.01 17 0.62 0.1 Timewell Drainage 7S Single-Wall HDPE

Knife Cut (Sand Slot)4 236 0.07 1.85 0.49 1.72 5.80 5.01 31 0.47 0.3

^{[a]}??is the perforation width, ?_{s }is the perforation length, ??_{p}is the valley width,_{v }cis the perforation spacing,Nis the number longitudinal rows ofperforations,Ris the outer radius of the pipe, and_{o }Ris the inner radius of the pipe (see fig. 1)._{i}

The Effective Radius of Sand-slot PipesSand-slot pipes are installed in soil with low clay and organic matter (i.e., non-cohesive or weakly cohesive soil) that is prone to sediment clogging of the drain pipes. Our results showed that the effective radius of 4-row sand-slot pipes varied from 0.1 to 0.7 cm (average 0.4 cm) for a wide range of perforation dimensions (table 3). The largest effective radius of any sand-slot pipe was 1.6 cm for the 8-row pipe (No. 4S), followed by 1.2 cm for the 6-row pipe (No. 2S), and the smallest average value was 0.4 cm for the 4-row pipes. This shows a trend of increasing effective radius from 0.4 to 1.6 cm with the increase in the number of longitudinal rows of perforations from 4 to 8 rows. Dierickx (1999) also showed that the most efficient method for increasing effective radius is increasing the number of longitudinal rows of perforations.

The Effect of Perforation Characteristics on the Effective Radius of the PipeTo demonstrate the effect of perforation length on effective radius, the perforation dimensions of the sand-slot pipe No. 6S were modified and its effective radius was calculated for two scenarios (fig. 4). In scenario A, the slot width was doubled from 0.05 to 0.10 cm at 0.005-cm increments while other perforation characteristics were kept constant. In scenario B, the slot length was doubled from 1.38 to 2.76 cm at 0.1-cm increments while other perforation characteristics were kept constant. The results showed that doubling the slot width from 0.05 to 0.10 cm resulted in doubling of the perforation area from 17 to 33 cm

^{2}/m, whereas effective radius had a minor increase from 0.12 to 0.18 cm (fig. 4). On the other hand, doubling the slot length from 1.38 to 2.76 cm resulted in doubling of the perforation area from 17 to 33 cm^{2}/m, while considerably increasing effective radius from 0.12 to 0.86 cm.While doubling the perforation width or length resulted in doubling of the perforation area, effective radius considerably increased only when increasing perforation length. The reason for the greater impact of perforation length is the reduction of flowline convergence towards the pipe openings when the openings cover more of the corrugation valleys (i.e., water travels a shorter distance to enter pipe openings). Therefore, increasing the perforation length is an effective method of increasing effective radius, thereby increasing water entry into the pipes. Increasing perforation width had a minimal effect on increasing flow into the drain pipe. Also, increasing perforation width, increases the risk of sediment clogging of the drain pipes in soil with a drain sedimentation problem. Dierickx (1999) also showed that increasing perforation length is the second most efficient method for increasing effective radius after increasing the number of longitudinal rows of perforations. Overall, the most efficient method for increasing effective radius (and drain inflow) is increasing the number of longitudinal rows of perforations, followed by increasing perforation length.

Figure 4. (left) Doubling perforation width of the 4-row sand-slot pipe No. 6S from 0.05 to 0.1 cm at 0.005-cm increments while other perforation characteristics were kept constant. (right) Doubling perforation length from 1.38 to 2.76 cm at 0.1-cm increments while other perforation characteristics were kept constant. The plot of perforation area versus effective radius for all the drain pipes showed a general trend of increasing effective radius with increasing perforation area (Supplementary Section S10). However, results showed that increasing perforation area does not necessarily increase effective radius. Therefore, perforation area alone cannot predict the effective radius of a pipe based on an empirical regression equation. Instead, appropriate analytical solutions from Dierickx (1999) should be used to calculate an effective radius that accounts for the perforation shape and pattern.

The Effective Radius of Sock-wrapped PipesTo prevent drain sedimentation, knitted-sock envelopes are used in non-cohesive and weakly cohesive soil because of their low structural strength (Stuyt et al., 2005; Vlotman et al., 2020). To calculate the effective radius of the sock-wrapped pipes for a wide range of soils, three saturated hydraulic conductivities (

K) were assumed for the soil surrounding the drain: 1.0 cm/h for fine sandy loam, 6.0 cm/h for a sandy loam, and 12.0 cm/h for fine sand (Vlotman et al., 2020)._{sd}Effective radius considerably increased from an average 0.6 cm to 5.9 cm after wrapping the regular-perforated pipes with a knitted-sock envelope (table 4). The increase in effective radius of the pipe was due to the reduction in entrance head loss. Mohammad and Skaggs (1983) also measured an increase in effective radius from 0.5 to 3.6 cm after surrounding a 100-mm diameter 6-row circular-perforated pipe with a 5-cm-thick gravel envelope. Sekendar (1984) reported an increase in effective radius from 0.5 to 2.5 cm after wrapping a 50-mm diameter 8-row perforated PVC pipe with synthetic fiber. Furthermore, the saturated hydraulic conductivity of the soil surrounding the drain pipe and pipe type had a minor effect on the effective radius of the sock-wrapped drain (Supplementary Section S5).

The effective radius of the sock-wrapped pipes ranged from 5.7 to 6.0 cm for all three soils and all pipe materials (table 4), which were close to the outer radius of the pipes. The 4-row regular-perforated 150-mm diameter sock-wrapped pipe had an effective radius of 8.7 cm, which was also close to its outer radius of 8.77 cm (Supplementary Section S6). Sock-wrapped pipes reduced the entrance head loss to near zero, so that the effective radius reached its maximum value (i.e., outer radius of the drain pipe). This means that the sock-wrapped pipes functioned as a completely open pipe without walls. The application of sock-wrapped pipes considerably reduces entrance head loss when installed in suitable soil, thereby improving water entry into the pipes. Furthermore, the pipe perforation type did not affect the effective radius of the sock-wrapped pipes (Supplementary Section S7).

Table 4. The effective radius of 100-mm diameter regular-perforated pipes wrapped with a knitted-sock envelope in three soils. Saturated Hydraulic Conductivity

of the surrounding soil,K_{sd}Pipe No. 1R 2R 3R 4R 5R 6R 7R 8R 9R 10R 11R 1.0 cm/h 5.9 6.0 6.0 6.0 6.0 6.0 6.0 6.0 6.0 5.9 5.9 6.0 cm/h 5.9 5.9 5.9 5.9 5.9 5.9 5.9 5.9 5.9 5.8 5.8 12.0 cm/h 5.8 5.8 5.9 5.9 5.9 5.9 5.9 5.9 5.9 5.8 5.7

Optimum Drain Spacing and Initial System Cost of Sock-wrapped and 4-row Sand-slot PipesTo determine the optimum drain spacing, an average effective radius of 0.4 and 5.9 cm was used for 4-row sand-slot pipes and sock-wrapped pipes, respectively. The results showed that sock-wrapped pipes had wider optimum drain spacing than 4-row sand-slot pipes across a wide range of lateral

Kvalues (2.0 to 6.0 cm/h) for all combinations of depth to restrictive layer and drain depths (fig. 5). The increase in drain spacing is due to the increase in effective radius (Stuyt and Dierickx, 2006). Therefore, the larger effective radius of the sock-wrapped pipes resulted in a wider optimum drain spacing than the 4-row sand-slot pipes._{e}

Figure 5. Optimum drain spacing and initial system cost of 4-row sand-slot ( R=0.4 cm) and sock-wrapped pipes (_{ef}R=5.9 cm) across a wide range of equivalent saturated hydraulic conductivities for soil with risk of sediment clogging of the drain pipes. The design drainage rate was calculated using the method described in Step 1 of the Methods section. Initial cost (material and installation) was $2.66/m for sand-slot pipes and $3.12/m for sock-wrapped pipes._{ef}The largest difference in the optimum drain spacing between the sock-wrapped and the 4-row sand-slot pipes occurred at the 350-cm depth to restrictive layer (fig. 5), which is in accordance with a previous study by Skaggs (1978) showing that the greatest impact of a drain envelope was in soil with deep profiles. Results of this study together with the previous study show that sock-wrapped pipes are more efficient at increasing drain spacing in deep soil profiles than they are in shallow soil profiles.

The initial system cost of sock-wrapped and 4-row sand-slot pipes varied across different combinations of factors (fig. 5). The combination of 350-cm depth to restrictive layer and 75-cm drain depth was the only one that resulted in lower initial cost for sock-wrapped pipes than for 4-row sand-slot pipes. This is because sock-wrapped pipes are more efficient at increasing drain spacing in deep soil profiles than in shallow soil profiles (see the previous paragraph). At

Kof 2.0 cm/h, sock-wrapped pipes were $7/ha cheaper than 4-row sand-slot pipes. As_{e}Kincreased, the difference between the initial system cost of sock-wrapped and 4-row sand-slot pipes converged until the cost difference reached zero at_{e}Kof 2.4 cm/h. Once_{e}Kincreased beyond 2.4 cm/h, the initial cost of the system with the 4-row sand-slot pipes became less than that of the sock-wrapped pipes. At_{e}Kof 6.0 cm/h, sock-wrapped pipes were $18/ha more expensive than 4-row sand-slot pipes. Sock-wrapped pipes can cost less than 4-row sand-slot pipes when installed at a shallow depth in a deep soil profile with a low_{e}Kvalue. Overall, the initial system cost of sock-wrapped pipes compared to 4-row sand-slot pipes primarily depends on site-specific conditions, sand-slot pipe material, and unit cost of the sock material._{e}

Optimum Drain Spacing and Initial System Cost of Sock-wrapped and 8-row Sand-slot PipesThe above comparative analysis was repeated for the 8-row sand-slot pipes (i.e., No. 4S with 1.6-cm effective radius) and sock-wrapped pipes (Supplementary Section S9). Results showed that even though the sock-wrapped pipe allowed for a wider optimum drain spacing than 8-row regular-perforated pipes for all combinations of factors, the 8-row sand-slot pipes were found to have a lower initial system cost than sock-wrapped pipes ($48/ha to $174/ha lower cost) in all combinations of factors (depth to restrictive layer, drain depths, and

Kvalues), when designed at the same design drainage rate and drain depth._{e}Even though 2020 pipe material and installation costs were used in the comparative analysis, the result of 8-row sand-slot pipes having a lower initial system cost than sock-wrapped pipes for all combinations of factors is still valid for 2022 market prices. As of 2022, the cost of 100-mm diameter pipe material was $1.48/m. The cost of 100-mm diameter pipe material with CARRIFF sock was $2.07/m. The cost of pipe installation (with or without sock) was $1.64/m. Based on these costs, the 8-row sand-slot pipes were found to have a lower initial system cost than sock-wrapped pipes ($63/ha to $234/ha lower cost) in all combinations of factors, when designed at the same design drainage rate and drain depth.

Comparison of the 2020 and 2022 data shows that the initial system cost difference between the 8-row sand-slot and sock-wrapped pipes increased from 2020 to 2022. This is because the unit material cost difference between the 8-row sand-slot and sock-wrapped pipe increased from $0.46/m in 2020 to $0.59/m in 2022. The unit material cost difference between these two pipes needs to be much lower for sock-wrapped pipes to provide a lower initial system cost than 8-row sand-slot pipes, when designed at the same design drainage rate and drain depth. The breakeven point for the sock-wrapped pipe to cost less than the 8-row sand-slot pipe is when the unit material cost difference ranges from $0.03/m to $0.23/m depending on the drainage design and soil conditions. Overall, 8-row sand-slot pipes had a lower initial system cost than sock-wrapped pipes in both 2020 and 2022 prices, while providing the same drainage performance (i.e., design drainage rate).

Optimum Drain Spacing and Initial System Cost of 8-row and 4-row Regular-perforated PipesTo determine the optimum drain spacing, an average effective radius of 0.6 and 1.9 cm were used for 4- and 8-row regular-perforated pipes, respectively. Results showed that 8-row regular-perforated pipes allowed for a wider optimum drain spacing than 4-row regular-perforated pipes across a wide range of

Kvalues (0.4 to 1.7 cm/h) for all combinations of depth to restrictive layer and drain depths (fig. 6). This was because of the larger effective radius of the 8-row regular-perforated pipes compared to the 4-row pipes. In a previous section, it was discussed that increasing the number of perforation rows considerably increases the effective radius of the pipe. The largest difference in optimum drain spacing between the 8-row regular-perforated pipes and the 4-row regular-perforated pipes occurred with the 350-cm depth to restrictive layer. This indicates that the use of 8-row pipes instead of 4-row pipes had the greatest impact on the optimum drain spacing in a deep soil profile._{e}For all four combinations of depth to restrictive layer and drain depths, the 8-row regular-perforated pipes resulted in a lower initial system cost than 4-row regular-perforated pipes across a wide range of

Kvalues. The difference in cost between the two pipe materials ranged from $11/ha to $227/ha for_{e}Kvalues ranging from 0.4 to 1.7 cm/h (fig. 6). The difference in the initial system cost between 8-row and 4-row regular-perforated pipes was smallest when the drain depth was at 125 cm (difference ranging from $11/ha to $33/ha). For pipes at a depth of 75 cm, the difference in the initial system cost between 8-row and 4-row regular-perforated pipes ranged from $78/ha to $227/ha. Even though 2020 pipe material and installation costs were used in this comparative analysis, the result of 8-row pipe having a lower initial system cost than 4-row pipes for all combination of factors, is valid under any market price because both pipes have the same unit material cost. The difference in cost between the two pipes originates from the wider drain spacing of the 8-row pipes than 4-row pipes to provide the same design drainage rate. Overall, the 8-row regular-perforated pipes had a lower initial system cost than the 4-row regular-perforated pipes for all combination of factors, when designed at the same design drainage rate and drain depth._{e}

ConclusionsThe theoretically estimated effective radii of 100-mm diameter pipes manufactured in the United States with various perforation characteristics, with and without a sock, resulted in the following key conclusions.

- For 3- or 4-row regular-perforated pipes, the effective radius ranged from 0.3 to 0.9 cm (average 0.6 cm). For 8-row regular-perforated pipes, the effective radius was 1.9 cm. For 4-row sand-slot pipes, the effective radius ranged from 0.1 to 0.7 cm (average 0.4 cm), whereas that of the 8-row sand-slot pipe was 1.6 cm.
- The most efficient method for increasing effective radius is increasing the number of longitudinal rows of perforations, followed by increasing perforation length. Increasing perforation width had a minimal effect on increasing flow into the drain pipe.
- The effective radius of sock-wrapped pipes ranged from 5.7 to 6.0 cm (average 5.9 cm), which functioned as a completely open pipe without walls.

Figure 6. Optimum drain spacing and initial system cost of 4-row ( R=0.6 cm) and 8-row (_{ef}R=1.9 cm) regular-perforated pipes across a range of equivalent saturated hydraulic conductivities for soil without risk of sediment clogging of the drain pipes. The design drainage rate was calculated using the method described in Step 1 of the Methods section. The initial cost (material and installation) for both pipes was $2.66/m._{ef}

- The larger effective radius of sock-wrapped pipes compared with that of sand-slot pipes, and that of 8-row regular-perforated pipes compared with that of 4-row regular-perforated pipes, resulted in a wider optimum drain spacing (ranging from 0.8 to 5.4 mwider), when designed at the same design drainage rate and drain depth.
- The initial system cost of sock-wrapped pipes compared to 4-row sand-slot pipes primarily depends on site-specific conditions, sand-slot pipe material, and unit cost of the sock material. Sock-wrapped pipes can have a lower initial system cost than 4-row sand-slot pipes when installed at a shallow depth in a deep soil profile with a low saturated hydraulic conductivity.
- The 8-row sand-slot pipes had a lower initial system cost than the sock-wrapped pipes in all combinations of factors, when designed at the same design drainage rate and drain depth. The breakeven point for the sock-wrapped pipe to cost less than the 8-row sand-slot pipe is when the unit material cost difference ranges from $0.03/m to $0.23/m depending on the drainage design and soil conditions.
- The 8-row regular-perforated pipes had a lower initial system cost than the 4-row regular-perforated pipes in soil without a drain sedimentation issue, when designed at the same design drainage rate and drain depth.
In conclusion, when there is no risk of drain sedimentation, 8-row regular-perforated pipes provided a lower initial system cost than 4-row regular-perforated pipes while providing the same drainage performance (i.e., design drainage rate). When drain sedimentation is a problem, 8-row sand-slot pipes provided a lower initial system cost than sock-wrapped pipes based on 2020 and 2022 prices, while providing the same drainage performance. The value of this study is that it provides effective radii of drain pipes required for drainage design and modeling, and informs drainage design decisions as to which pipe material is more suitable for site-specific conditions.

AcknowledgementsThe author expresses gratitude to Yousef AbdalAal for measuring the perforation dimensions. The author thanks the drainage manufacturers for donating samples of their drain pipes. Funding of this research was provided by Michigan State University (faculty startup funds).

Supplemental MaterialsSupplementary Materials associated with this article can be found at FigShare (https://figshare.com/s/489ae7879278ba0accd1).

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