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2. CHAPTER 3 SUB-SURFACE DRAINAGE THEORY ERNST EQUATION 2

3. 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. 3

4. ERNST EQUATION The total head loss in the system (H) is sum of the head losses due to the vertical flow (H v ), the head loss during horizontal flow (H h ) and head loss during the radial flow (H r ) 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. 4

5. 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.  D v = distance over which vertical flow takes place.  D r = distance over which radial flow takes place. 5

6. Flow geometry for pipe drain located in lower layer of a two layer soil 6

7. Geometry for pipe drain located in upper layer of a two layer soil. 7

8. Nomenclature used in Figures  D 1 =average thickness of upper layer below the water table,  D 2 =thickness of the lower layer,  D 0 =average thickness of the layer in which drain is located below drain level,  y =water depth in ditch drain (for pipe drain y = 0). 8

9. Vertical Flow  The vertical flow distance, D v, can be taken as y+H or y+H/2 for ditches and as H or H/2 for pipe drainage.  The vertical flow distance and, therefore, head loss in vertical flow, is small in magnitude in comparison with horizontal or radial flow.  Thus use of H or H/2 will not make any significant difference. 9

10. Horizontal Flow  The horizontal flow takes place over a distance S and through section of thickness D 1 +D 2.  The horizontal conductance is given as:  For a very deep stratum the horizontal conductance will be a very large quantity, and H h can become very small by use of Eq. (3.42), indicating a very large drain spacing.  This is practically not valid, necessitating an upper limit as: D1 < S/4 and D 1 +D 2 ≤ S/4. 10

11. Radial Flow  The radial flow is assumed to be occurring only in the layer immediately below drain level.  Thus the distance over which radial flow takes place is taken equal To the depth below the drain to the interface (if interface is below drains) or To the lower impermeable layer (if interface is above the drain), thus D r = D 0 with D 0 ≤ S/4 to avoid a very small head loss in radial flow. 11

12. Geometry factor (a) and application of Ernst's Equation  The geometry factor (a) is determined considering the relative positions of the pipe drain viz-a-viz the soil layers and the conductivity ratios of these layers. Different cases are identified as follows. Case-I: Homogeneous Soil D 2 = 0, D v = y+H/2, K r = K 1, D r = D 0, and a = 1. Case-II: Layered Soil For the case of layered soil the pipe drain may be located in the upper layer or lower layer. Case II a: Pipe drain located in lower layer of a two layer soil  When the pipe drain is located in lower layer of a two layer soils (Fig. 3.10), then two situations can arise. i) K 1 K 2. 12

13. Geometry factor (a) and application of Ernst's Equation For the first case of K 1 < K 2, the geometry factor is taken as: a = 1, and following explanation will be applicable. a. Vertical resistance is considered only in upper layer of lesser hydraulic conductivity (K1) and D v = H/2. The vertical head loss in lower layer is insignificant and ignored. b. Horizontal conductance is given as: Σ(KD)h = K 1 D 1 + K 2 D 2. But K 1 D 1 may be neglected being of small value (D 1 << H). c. Radial flow is in lower layer and, therefore, D r = D 0 and K r = K 2 with the limitation of D 0 ≤ S/4. 13

14. Geometry factor (a) and application of Ernst's Equation Case II b: Pipe drain located in upper layer of a two layer soil. The selection of the geometry factor depends upon the ratio of the hydraulic conductivity of the two layers (Fig. 3.11). Three situations are identified as: (i)K 2 > 20 K 1 The geometry factor is taken as : a = 4, D v = H/2, Σ(KD) h = K 1 D 1 + K 2 D 2, D r = D 0, K r = K v = K 1 (ii) 0.1 K 1 < K 2 < 20 K 1 In this case the geometry factor (a) is obtained from nomograph of Fig. 3.12 or Table 3.3 on the basis of ratio of hydraulic conductivity of upper and lower layers (K2/K1) and the ratio of thickness of lower layer to radial flow distance (D2/D0). 14

15. Nomograph for case (0.1 K 1 < K 2 < 20 K 1 ) 15

16. Table for case (0.1 K 1 < K 2 < 20 K 1 ) 16

17. Example 3.3 A soil consists of two layers. The upper layer is 4.5 m (K 1 = 0.6 m/d) and lower layer is 3.5 m thick (K 2 = 2.6 m/d). The pipe drains of 15 cm dia are installed at a depth of 2.5 m below ground surface. The required root zone depth is 1.5 m. The constant recharge rate is 2.5 mm/d. Calculate the drain spacing. 17

18. Solution of Example 3.3. 18

19. DONAN EQUATION USBR had used Donan Formula for determining pipe drain spacing. The Donan formula is given as (Fig. 3.1): where the variables are as defined earlier. 19

20. Modified Donan Equation  The modified Donan equation was used to calculate drain spacing in Fourth Drainage Project (Faisalabad) by USBR team (USBR, 1989).  The modified Donan formula is applicable for a two layer soil with hydraulic conductivity and depth of lower layer greater than the upper layer.  It is considered that all recharge flows vertically through top layer of lower hydraulic conductivity to lower layer of higher hydraulic conductivity.  The horizontal flow is considered to be restricted with the lower layer due to smaller flow resistance because of higher hydraulic conductivity. When the flow reaches closer to the pipe drain, it travels back to the drain. 20

21. Symbols for Modified Donan equation. 21

22. Modified Donan Equation  Thus the total head loss is determined as: Where H 1 =Head loss for vertical flow from top of water table to the interface of the layers. H 2 =Head loss for horizontal flow in layer 2. H 3 =Head loss for radial flow from layer 2 back to the drain. The head loss for vertical flow (H 1 ) is given as: 22

23. Modified Donan Equation  The head loss for radial flow (H 3 ) is determined as: Where  The head loss for horizontal flow (H 2 ) is determined from Eq. 3.44  The drain spacing, S, is computed using H 2 as: 23

24. Modified Donan Equation Since spacing S is required for determining H 3 and H 2, direct solution is not available. On solving Eq. (3.46) for S, and squaring and then equating it with Eq. (3.48), a quadratic equation results of the form: 24