2. CHAPTER 3 SUB-SURFACE DRAINAGE THEORY

3. SUB-SURFACE DRAINAGE THEORY
The ground watertable is required to be maintained beyond the root zone depth, Y, for successful crop growth.
This is achieved by providing sub-surface drainage usually in the form of pipes buried below the ground surface.
The sub-surface drainage is interchangeably also referred as pipe drainage.

4. Nomenclature and symbols for a sub-surface pipe drainage system.

5. Sub-Surface Drainage
The sub-surface pipe drains are installed at a depth, W, below the ground surface to keep the ground water table at the desired depth, Y.
The pipes are usually installed parallel to each other at a constant spacing of S.
The pipes are of radius r and are laid at some slope towards pipe‟s discharge end.
The recharge, R, from above the ground surface is intercepted by the drains and is carried away through the pipe.
In some cases deep open ditch drains may be constructed. The pipe or open ditch drains may or may not reach the lower impermeable layer.

6. Recharge Condition Steady State Condition Unsteady state condition
When the recharge is constant, the water table stays at a constant position. This type of condition is referred as steady state condition.
This situation can occur when rainfall of relatively small intensity continues for a long time (theoretically infinite time).
if the recharge is large, as due to heavy rainfall, and recharge events are sporadic, this may result in rising of water table. The water table will fall back during the succeeding period when no recharge occurs. This condition is referred as unsteady state or transient condition
Apart from heavy rainfall, unsteady state condition can also arise due to periodic irrigation applications which are usually applied after seven to ten days interval.

7. Sub-surface flow to parallel and fully penetrating ditch drains.
THEORY OF GROUND WATER FLOW TO DRAINS

8. THEORY OF GROUND WATER FLOW TO DRAINS
At any point in the flow domain, say at the section located a distance X from the drain edge, the components of the flux density (q) are given by the Darcy's law as
…..(3.1)
The resultant flux density in the actual flow direction, ℓ, is given as:
………….(3.2)
………….(3.3)

9. Dupuit - Forchheimer (D-F) Approximations
The D-F approximations / simplifications states that:
For small inclinations of the water surface the streamlines or flow lines can be taken as horizontal; and
The flux density/ velocities associated with these inclinations are proportional to the slope of the free water surface and is independent of the depth

10. STEADY STATE DRAINAGE DESIGN
The steady state drainage design is based upon the assumption that a steady and constant recharge occurs through the soil to the drains.
The discharge into the drains and the water table position is constant in time.
Different methods used for steady state drainage design and analysis include Hooghoudt, Kirkham, Youngs, Ernst, Donan, Modified Donan, Van Beers, etc and few of these are discussed below.

11. HOOGHOUDT EQUATION
At any vertical section in the flow domain (X = x), the flow passing through the section equals to all recharge entering in the profile to the right of the section, i.e. through the distance S/2-x. ……………….(3.4) Comparing Eq. (3.3) and (3.4) ………………...(3.5) Or …………………(3.6)

12. HOOGHOUDT EQUATION
Integrating equation 3.6 and Applying Limits When x=0, h=D and x=S/2, h=hm=H+D …………(3.7) Therefore, ………………(3.10)

13. Parameters of Hooghoudt Equation
D = Water depth in the ditch drain above impermeable layer (L);
H = height of watertable midway between the drains above drain water level (L);
h = height of watertable above drain water level at a distance x from the drain (L);
hm= height of watertable midway between the adjacent drains above impermeable bed (L). (hm = D + H);
K = saturated hydraulic conductivity of the soil (L/T),
q = drain discharge per unit surface area (L/T) (q = R for steady state conditions);
R = constant recharge rate per unit surface area (L/T); and
S = drain spacing between the adjacent drains (L).

14. HOOGHOUDT EQUATION
Again Integrating equation 3.6 and Applying Limits When x=0, h=D and x=x, h=h, ……………..(3.11) …………..(3.12) The Eq. (3.12) is equation of an ellipse and hence the water surface profile developed in sub-surface drainage is section of an ellipse.

15. HOOGHOUDT EQUATION
The Eq. (3.10) may be written as: For, hm=H+D, Where H = height of water table midway between the drains above the drain water level), then: The Eq. (3.15) is another form of Hooghoudt Equation and can be used to determine the drain spacing when the hydro-geologic characteristics (i.e. K, R, D, H) of the area are known.

16. HOOGHOUDT EQUATION
If D = 0, i.e. no water in the drain, then Eq. (3.15) becomes as:
Eq. (3.16) represents the horizontal flow occurring in the part of the profile above the drain level.
If the depth D is large compared to H, then
Eq. (3.17) represents the horizontal flow occurring in the part of the profile below the drain level.

17. HOOGHOUDT EQUATION
If the profile consists of two distinct layers of different hydraulic conductivity with interface located at the drain water level, then Eq. (3.15) may be written as:
…………..(3.18)
Where Ka and Kb is the hydraulic conductivity of the layers above and below the drain water level, respectively.

18. Flow Lines

19. Flow lines
The recharge R entering into the saturated soil profile ultimately discharge into the ditch drain through its side.
In doing so the flow lines are oriented vertically at the water surface and are oriented almost horizontally at the exit face (Fig. 3.4).
The flow lines are essentially curved between the entrance and exit faces as shown in Fig. 3.4 (a).
The flow lines in ditch drainage are thus not completely horizontal for their full length. Actual flow lines are horizontal only at a local (or small) scale and are curvilinear at global (or large) scale.
The flow lines are simplified as vertical for the first part and horizontal for the later part as shown in Fig. 3.4 (b).

20. Partial Penetration of Drains
Subsurface drains mostly consist of pipe drain of certain radius, r0, and placed at some distance D above the lower impermeable barrier (Fig. 3.1).
If subsurface drains are open ditch drains, these ditch drains also seldom reach the barrier.
Thus the pipe or ditch drains only partially penetrate the soil profile.
Due to the partial penetration of pipe drains or ditch drains the flow lines do not follow the simplified orientation shown in Fig. 3.4 (b) but are oriented as Fig. 3.5 (a).
The flow region is divided into a short vertical flow region, a horizontal flow region and a radial flow region.
This involves additional head loss in radial flow region.
It is considered that flow will be radial in a region of radius D around the pipe drain.

21. Flow Lines for Partially penetrating Ditch Drains

22. HOOGHOUDT EQUATION FOR PARTIALLY PENETRATING DRAINS
The Hooghoudt's equation can in general be written as: where FH is Hooghoudt's resistance factor and is given as: The first term on RHS of Eq. (3.20) represents the head loss due to horizontal flow. The last two terms represent the head loss due to radial flow.

23. HOOGHOUDT EQUATION FOR PARTIALLY PENETRATING DRAINS
Hooghoudt postulated that a pipe drain or a partially penetrating ditch drain placed a height D above the impermeable barrier can be conceptually replaced by an equivalent fully penetrating ditch drain for which the impermeable barrier is at a depth d and is shallow than the actual depth D such that d < D.
The depth d is termed as equivalent depth, d, and represents the depth to equivalent (but fictitious) impermeable layer below the pipe drain or the ditch drain.
Thus the actual depth D in Eqs. (3.10), (3.15) and (3.18) is replaced by the equivalent depth d to account for extra radial resistance encountered during the flow towards the pipe drains or partially penetrating ditch drains.

24. HOOGHOUDT EQUATION FOR PARTIALLY PENETRATING DRAINS
Equation 3.10 becomes as
Equation 3.15 becomes as
Equation 3.18 for two layer soils becomes as
The equivalent depth (d) is function of spacing (S), drain radius (r), and depth to the impermeable layer (D).

25. How to Determine Equivalent Depth (d)

26. How to Determine Equivalent Depth (d)

27. SOLUTION OF HOOGHOUDT EQUATION
For a fully penetrating ditch drain the drain spacing (S) is determined by using Eq. (3.15) or (3.18).
For pipe drains or partially penetrating ditch drains, equivalent depth (d) is used in place of actual depth (D) of the profile.
The equivalent depth itself is dependent upon the drain spacing (S).
Thus solution of the Hooghoudt's equation for determining drain spacing (S) becomes more complex.
The spacing (S) is obtained by a trial-and-error method.
Alternatively nomograph (Fig. 3.7) may be used for direct and quick solution.

28. EXAMPLE 3.1 FOR DRAIN SPACING
Determine pipe drain spacing for Swabi SCARP soils by using Hooghoudt method. H = 0.8 m, K = 1.56 m/d, dia = 15 cm, R = 2 mm/d, D = 5 m. Use trial and error procedure.
SOLUTION
-The spacing is determined by using Eq. (3.22).
-Using given data in the equation:
S2 = [8 × 1.56 × 0.8 × d + 4 × 1.56 × 0.82] / 0.002
= 4992 d
or S = (4992 d )0.5
Trial 1: let d = D = 5 m. Then S = (4992 × )0.5 = 164 m
Trial 2: For S = 164 m, D = 5 m, d (from Table 3.1) = Then
S = (4992 × )0.5 = 150 m
Trial 3: For S = 150 m, D = 5 m, d (from Table 3.1) = Then
S = (4992 × )0.5 = m.
Therefore adopted drain spacing = 150 m.

29. KIRKHAM SOLUTION
Kirkham (1958) based the solution for the pipe drain spacing on the exact solution of the potential theory.
The flux at any point in the flow domain is given by Eq. (3.2). Using Eq. (3.1) in Eq. (3.2) and noting that for a steady system there is no change in the flux, i.e. dq = 0.
Therefore
…………….(3.31)
Kirkham obtained the solution of the Eq. (3.30) in terms of H as:

30. KIRKHAM SOLUTION
In deriving Eq. (3.31) Kirkham considered the flow as only vertical in the region above drain level with very negligible head loss and as only horizontal flow in the region below the drain level. The Eq. (3.31) may be written as:
………….(3.32)
where FK is Kirkham's correction factor given as:

31. How to find Fk

32. KIRKHAM SOLUTION
The Kirkham's solution for two layer soil (with interface at drain level) is given as:

33. Example 3.2
Determine pipe drain spacing by Kirkham Method for Swabi SCARP soils. H = 0.8 m, K = 1.56 m/d, dia = 15 cm, R = 2 mm/d, D = 5 m.
Solution
The solution is obtained by trial and error as under.
- D/2r = 5/(2 x 0.075) = 33.33
-Let S = 100; S/D = 100/5 = 20. From Table 3.2 FK = 3.8 (after interpolation), thus H = (0.002 × 100 × 3.8) / 1.56 = 0.48 ≠ 0.8. (rejected) -Let S = 120; S/D = 120/5 = 24. From Table 3.2 FK = 4.08 (after interpolation), thus H = (0.002 × 120 × 4.08) / 1.56 = 0.63 ≠ 0.8 (rejected)
-Let S = 140; S/D = 140/5 = 28. From Table 3.2 FK = 4.20 (after interpolation), thus H = (0.002 × 140 × 4.20) / 1.56 = 0.75 ≠ 0.8 (rejected)
-Let S = 145; S/D = 145/5 = 29. From Table 3.2 FK = 4.22 (after interpolation), thus H = (0.002 × 145 × 4.22) / 1.56 = 0.78 ≠ 0.8 (rejected)
-Let S = 150; S/D = 145/5 = 30. From Table 3.2 FK = 4.25 (after interpolation), thus H = (0.002 × 150 × 4.25) / 1.56 = 0.81 ≈ 0.8 (accepted)
-Therefore selected S = 150 m

34. ERNST EQUATION
The drainage equations discussed earlier have limitations that these can be used for two layer soils only when the drain is located precisely at the interface of the two layers.
In practice this situation seldom arises as the interface is usually not same depth below ground surface everywhere and that drains are laid at some slope.
The Ernst's equation is applicable to two layer soil with interface at any place.
It has further favorable point that the head loss during the vertical flow is also included.

35. ERNST EQUATION
The total head loss in the system (H) is sum of the head losses due to the vertical flow (Hv), the head loss during horizontal flow (Hh) and head loss during the radial flow (Hr) in different parts of the flow domain.
In general the total head loss (H) is given as:
The flow of water in porous medium, given by Darcy's law, is comparable to flow of electricity given by Ohm's laws written as: V = I R. Similarly the flow of water can be written as: H = q W, where q is the flow rate and W is the resistance to the flow.

36. ERNST EQUATION The Eq. (3.39) can accordingly be written as: OR where:
a = geometry factor for radial flow depending upon the flow conditions and is to be determined in relation to soil profile and drain layout.
Dv = distance over which vertical flow takes place.
Dr = distance over which radial flow takes place.

37. Flow geometry for pipe drain located in lower layer of a two layer soil

38. Geometry for pipe drain located in upper layer of a two layer soil.