Finite element modelling of entrance resistance for perforated

HDPE subsurface drainage pipes

N. Gaj and Prof. C.A. MadramootooDept. of Bioresource Engineering

McGill University, Montreal07 Sept, 20181

CSBE/SCGAB Annual General Meeting and Technical ConferenceGuelph, 2018

BackgroundSubsurface drainage materials

High density polyethylene (HDPE)has been used since the 1950s.

Benefits:Economically feasibleEasy to handle and transportMechanized installation

2

Clay tiles

HDPE tubing

Background

Drainage theories for subsurface pipe design assume pipe walls are completely porous.

Field installations have finite openings or perforations on the pipe wall, causing additional hydraulic head losses known as the entrance resistance, αe.

Previous evaluations of αe have been based on electrical analogue models and have not been adequately validated for corrugated HDPE pipes.

3

From Dierickx, 1980

( )

−=

0

120

ln

2

RR

LkQ T φφπ

Background

Theory for radial planar flow and entrance resistance

4

From Dierickx, 1980

Research objectivesTo develop general guidelines for perforation design in HDPE pipes used in agricultural land drainage.

to assess the effects of perforations on the flow towards the drainpipes.a) establish the zone of influence for quasi-radial flow convergence towards perforationsb) establish the effect of perforation shape, size, and pattern on the entrance resistance and delivery ratio (Q/Q0)

5

Research objectives

to assess the effects of perforations on the local stresses in the walls of the drainpipes.

a) establish the effects from agro-traffic, embedment depth, and soil type on local stress concentration factors (λSC) around perforations

b) establish the effect of perforation shape, size, and pattern on the stresses (σc) and deformation (ΔD) in the pipe wall

6

Research objectives

to establish design guidelines for perorations on HDPE drainpipes

a) generate design charts for the effective depth (de) vs. drain spacing (L) based on λSC and Q/Q0for Eastern Canada.

7

Materials and Method

Numerical Simulations

and Analyses

•Computation of entrance resistance, αe

•Assessment of αe for various perforation shape, size , and pattern

Numerical Modelling

•Geometry and parameter set-up

•Finite Element mesh assessment

•Model Calibration

Sand tank experiments

•Characterizing porous medium

•Flow rate and piezometric head measurements

8

Materials and Method

Porous medium characteristics

Property Value

Void ratio, e 0.44 to 0.83

Effective size, D10 0.34 mm

Coefficient of Uniformity, Cu 2.1

Specific Gravity, Gs 2.662

Max. Dry Density, γd max 15.62 kN/m3

Optimum Moisture Content, OMC 7.7%

Hydraulic Conductivity, Ksat (Kozeny-Carman model) 53.9 to 247.1 m/d

9

Materials and Method

Sand tank experiments

10

Rainfall simulators

Supply reservoir

Rectangular flume

Gated-outlet control chamber

Materials and Method

Sand tank experiments

11

Model PC:Plain Wall and Circular Holes

Model CC:Corrugated

Wall and Circular Holes

Model CS:Corrugated

Wall and Slits

Materials and Method

Perforation pattern

12

Materials and Method

Perforation pattern

13

Materials and Method

Perforation pattern

14

Materials and Method

Instrumentation and sensors

15

CR23X data logger

V-notch discharge weir

(11° angle)

Piezometers (10 mm dia.)

Sonic sensor (SR 50)

Collection reservoir

Materials and Method

Electrical analogue approach (Dierickx, 1980)

where:αe is the entrance resistance, m is the number of perforations per unit drain length, λ1 and λ2 are the smallest and largest perforation spacing respectively (m), δp is the diameter of the perforation (m).

16

−−=

1

2

1

ln291.32

111λλ

λδπα

pea m

- arched boundary conditions on PC model

−−=

1

2

1

ln291.32

12

1λλ

λδπ

πα

pep m

- plane boundary conditions on PC model

Materials and Method

Numerical modelling

17

Model geometry and set-up

Mesh generation for

model

Materials and Method

Numerical modelling – mesh assessment (PC)

18

Variation of model output with increasing

number of elements

Mesh quality for model PC

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

1.6E-04

- 50,000 100,000 150,000 200,000 250,000

Mod

el O

uput

Number of Elements

Materials and Method

Numerical modelling – mesh assessment (CC)

19

Variation of model output with increasing

number of elements

Mesh quality for model CC

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

Mod

el O

uput

Number of Elements

Preliminary Results

Numerical modelling (Calibration -PC)

20

Ks1 Ks2 Ks3 Ks4 Ks1 Ks2 Ks3 Ks4

cm m3/s 1.530E-03 1.560E-03 1.590E-03 1.620E-03 1.530E-03 1.560E-03 1.590E-03 1.620E-0340.1 1.4088E-04 1.357E-04 1.384E-04 1.410E-04 1.437E-04 5.1532E-06 2.4919E-06 1.6938E-07 2.8307E-0645.2 1.3979E-04 1.208E-04 1.232E-04 1.255E-04 1.279E-04 1.8982E-05 1.6613E-05 1.4244E-05 1.1875E-0550.1 1.0439E-04 1.065E-04 1.086E-04 1.106E-04 1.127E-04 2.0868E-06 4.1746E-06 6.2623E-06 8.3500E-0655.1 8.8603E-05 9.185E-05 9.365E-05 9.545E-05 9.725E-05 3.2462E-06 5.0471E-06 6.8481E-06 8.6490E-06

MAE 7.3669E-06 7.0816E-06 6.8809E-06 7.9262E-06

Simulated Discharge, Qs Absolute Error (AE)Measured

discharge, Qm

Hydraulic head on the

discharge pipe, Hd

Preliminary Results

21

Hd

Hydraulic head on outlet from

bottom of pipe

Q αe

cm cm m3/sPipe empty PC-0 27.375 0 2.36E-04 0.387373

1/4 full PC-0.25 30.2375 2.863 2.25E-04 0.4147011/2 full PC-0.5 33.1 5.725 2.14E-04 0.4449653/4 full PC-0.75 35.9625 8.588 2.02E-04 0.478651

Full PC-F 38.825 11.450 1.91E-04 0.516381PC-1 40.1 12.725 1.86E-04 0.534697PC-2 45.2 17.825 1.65E-04 0.619217PC-3 50.1 22.725 1.46E-04 0.722728PC-4 55.1 27.725 1.26E-04 0.861677

From Sand Tank

Experiments

Condition on outlet

Code

Preliminary Results

Numerical modelling - convergence of streamlines - PC

22

Plan view

Front view

Preliminary Results

Numerical modelling - convergence of streamlines - CC

23

Plan view

Front view

Preliminary Results

Numerical simulations - convergence of streamlines with depth

24Front view

Preliminary Results

Numerical simulations - convergence of streamlines with width

25Front view

Preliminary Results

26

Effects of the tank width, W and depth, de on the flow rate through a fully porous wall

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

0 50 100 150 200 250 300 350 400

Flow

rate

, Q0

(m3 /

s)

Width of tank, W (cm)

30.6 cm

50.6 cm

70.6 cm

90.6 cm

110.6 cm

Q0 = 6.44 E-04

Preliminary Results

27

Effects of the tank width, W and depth, de on the flow rate through perforations for model PC

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

1.6E-04

1.8E-04

2.0E-04

0 50 100 150 200 250 300 350 400

Flow

rate

, Q (m

3 /s)

Width of tank, W (cm)

30.6 cm

50.6 cm

70.6 cm

90.6 cm

110.6 cm

Preliminary Results

28

Effects of the tank width, W and depth, de on the delivery ratio, Q/Q0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 50 100 150 200 250 300 350 400

Del

iver

y Ra

tio, Q

/Q0

Width of tank, W (cm

30.6 cm

50.6 cm

70.6 cm

90.6 cm

110.6 cm

Summary of FindingsEntrance resistances depend on the outlet condition of thedrainpipe.

The radial zone is not affected by the depth of theimpermeable layer.

The radial zone extends to 1.5 m laterally from the centre of thedrainpipe.

The benchmark model for Q0 comes within 0.7% error ofthe analytical solution for radial planar flow.

A delivery ratio of 0.28 was computed for the PC model

29

Next steps

Continue the simulations and analyses for thecorrugated pipe models

Assess the effects of perforations on the local stressesin the pipe wall.

Amalgamate the hydraulic and structural designcomponents to develop general design guidelines forvarious field conditions in Eastern Canada.

30

References

COMSOL Multiphysics. (2018). Subsurface Flow Module:User's Guide. USA: COMSOL.

Dierickx, W. (1980). Electrolytic analogue study of the effectof openings and surrounds of various permeabilities onthe performance of field drainage pipes. Doctoral thesis,Wageningen, Merelbeke, Belgium.

Skaggs, R. W. (1978). Effect of Drain Tube Opening on Water-Tabl e Drawdown. Journal of Irrigation and DrainageDivision, (ASCE), 104(IR1), 13-21.

31

Questions? Thank You!

32Please vote at NSERC’s Science Exposed 2018 Image competition

Finite element modelling of entrance resistance for perforated HDPE subsurface drainage pipesBackgroundBackgroundBackgroundResearch objectivesResearch objectivesResearch objectivesMaterials and MethodMaterials and MethodMaterials and MethodMaterials and MethodMaterials and MethodMaterials and MethodMaterials and MethodMaterials and MethodMaterials and MethodMaterials and MethodMaterials and MethodMaterials and MethodPreliminary ResultsPreliminary ResultsPreliminary ResultsPreliminary ResultsPreliminary ResultsPreliminary ResultsPreliminary ResultsPreliminary ResultsPreliminary ResultsSummary of FindingsNext stepsReferencesQuestions? Thank You!