Flow nets

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Slide 1: 

Flow Nets

Slide 2: 

Flow through a Dam Drainage blanket Phreatic line Unsaturated Soil Flow of water z x

Slide 3: 

Graphical representation of solution 1. Equipotentials Lines of constant head, h(x,z) Equipotential (EP)

Slide 4: 

Phreatic line Flow line (FL) 2. Flow lines Paths followed by water particles - tangential to flow Graphical representation of solution Equipotential (EP)

Slide 5: 

Properties of Equipotentials h(x,z) = constant (1a) Flow line (FL) Equipotential (EP)

Slide 6: 

h(x,z) = constant (1a) Thus: (1b) Properties of Equipotentials Flow line (FL) Equipotential (EP)

Slide 7: 

h(x,z) = constant (1a) Thus: (1b) Equipotenial slope (1c) Properties of Equipotentials Flow line (FL) Equipotential (EP)

Slide 8: 

Geometry vz vx Kinematics Properties of Flow Lines From the geometry (2b) Flow line (FL) Equipotential (EP)

Slide 9: 

Geometry vz vx Kinematics Properties of Flow Lines From the geometry (2b) Now from Darcy’s law Flow line (FL) Equipotential (EP)

Slide 10: 

Geometry vz vx Kinematics Properties of Flow Lines From the geometry (2b) Now from Darcy’s law Hence (2c) Flow line (FL) Equipotential (EP)

Slide 11: 

Orthogonality of flow and equipotential lines On an equipotential On a flow line Flow line (FL) Equipotential (EP)

Slide 12: 

Orthogonality of flow and equipotential lines On an equipotential On a flow line Hence (3) Flow line (FL) Equipotential (EP)

Slide 13: 

X y z t T Y Z X FL FL Geometric properties of flow nets h h+h h+2h EP

Slide 14: 

X y z t T Y Z X FL FL (4a) From the definition of flow Geometric properties of flow nets h h+h h+2h EP

Slide 15: 

X y z t T Y Z X FL FL (4a) (4b) From the definition of flow From Darcy’s law Geometric properties of flow nets h h+h h+2h EP

Slide 16: 

X y z t T Y Z X FL FL (4a) (4b) (4c) From the definition of flow From Darcy’s law Combining (4a)&(4b) Geometric properties of flow nets h h+h h+2h EP

Slide 17: 

X y z t T Y Z X FL FL (4a) (4b) (4c) (4d) From the definition of flow From Darcy’s law Combining (4a)&(4b) Similarly Geometric properties of flow nets h h+h h+2h EP

Slide 18: 

X y z t T Y Z X FL FL (4a) (4b) (4c) (4d) From the definition of flow From Darcy’s law Combining (4a)&(4b) Similarly Geometric properties of flow nets h h+h h+2h EP

Slide 19: 

a b c d D B C A Geometric properties of flow nets FL EP( h ) EP ( h + h )

Slide 20: 

(6a) From the definition of flow a b c d D B C A Geometric properties of flow nets FL EP( h ) EP ( h + h )

Slide 21: 

(6a) (6b) From the definition of flow From Darcy’s law a b c d D B C A Geometric properties of flow nets FL EP( h ) EP ( h + h )

Slide 22: 

(6a) (6b) (6c) (6d) From the definition of flow From Darcy’s law Similarly Combining (6a)&(6b) a b c d D B C A Geometric properties of flow nets FL EP( h ) EP ( h + h )

Slide 23: 

(6a) (6b) (6c) (6d) From the definition of flow From Darcy’s law Similarly Combining (6a)&(6b) Conclusion a b c d D B C A Geometric properties of flow nets FL EP( h ) EP ( h + h )

Slide 24: 

When drawing flow nets by hand it is most convenient to draw them so that Each flow tube carries the same flow Q The head drop between adjacent EPs, h, is the same Then the flow net is comprised of “SQUARES” Geometric properties of flow nets

Slide 25: 

Geometric properties of flow nets Demonstration of ‘square’ rectangles with inscribed circles

Slide 26: 

Drawing Flow Nets To calculate the flow and pore pressures in the ground a flow net must be drawn. The flow net must be comprised of a family of orthogonal lines (preferably defining a square mesh) that also satisfy the boundary conditions.

Slide 27: 

Water Datum H-z z H (7) Common boundary conditions a. Submerged soil boundary - Equipotential h u z w w   

Slide 28: 

Water Datum H-z z H (7) Common boundary conditions a. Submerged soil boundary - Equipotential h u z now u H z w w w w       ( )

Slide 29: 

Water Datum H-z z H (7) Common boundary conditions a. Submerged soil boundary - Equipotential h u z now u H z so h H z z H w w w w w w             ( ) ( )

Slide 30: 

Permeable Soil Flow Line vn=0 vt Impermeable Material Common boundary conditions b. Impermeable soil boundary - Flow Line

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Common boundary conditions c. Line of constant pore pressure - eg. phreatic surface Head is given by

Slide 32: 

Common boundary conditions c. Line of constant pore pressure - eg. phreatic surface Head is given by and thus

Slide 33: 

Common boundary conditions c. Line of constant pore pressure - eg. phreatic surface Head is given by and thus now if pore pressure is constant

Slide 34: 

Common boundary conditions c. Line of constant pore pressure - eg. phreatic surface Head is given by and thus now if pore pressure is constant and hence (8)

Slide 35: 

Common boundary conditions c. Line of constant pore pressure - eg. phreatic surface

Procedure for drawing flow nets : 

Procedure for drawing flow nets Mark all boundary conditions Draw a coarse net which is consistent with the boundary conditions and which has orthogonal equipotentials and flow lines. (It is usually easier to visualise the pattern of flow so start by drawing the flow lines). Modify the mesh so that it meets the conditions outlined above and so that rectangles between adjacent flow lines and equipotentials are square. Refine the flow net by repeating the previous step.

Slide 37: 

Value of head on equipotentials Phreatic line (9) Datum 15 m h = 15m h = 12m h = 9m h = 6m h = 3m h = 0

Slide 38: 

For a single Flow tube of width 1m: Q = k h (10a) Calculation of flow Phreatic line 15 m h = 15m h =12m h = 9m h = 6m h = 3m h = 0

Slide 39: 

For a single Flow tube of width 1m: Q = k h (10a) For k = 10-5 m/s and a width of 1m Q = 10-5 x 3 m3/sec/m (10b) Calculation of flow Phreatic line 15 m h = 15m h =12m h = 9m h = 6m h = 3m h = 0

Slide 40: 

For a single Flow tube of width 1m: Q = k h (10a) For k = 10-5 m/s and a width of 1m Q = 10-5 x 3 m3/sec/m (10b) For 5 such flow tubes Q = 5 x 10-5 x 3 m3/sec/m (10c) Calculation of flow Phreatic line 15 m h = 15m h =12m h = 9m h = 6m h = 3m h = 0

Slide 41: 

For a single Flow tube of width 1m: Q = k h (10a) For k = 10-5 m/s and a width of 1m Q = 10-5 x 3 m3/sec/m (10b) For 5 such flow tubes Q = 5 x 10-5 x 3 m3/sec/m (10c) For a 25m wide dam Q = 25 x 5 x 10-5 x 3 m3/sec (10d) Calculation of flow Phreatic line 15 m h = 15m h =12m h = 9m h = 6m h = 3m h = 0

Slide 42: 

For a single Flow tube of width 1m: Q = k h (10a) For k = 10-5 m/s and a width of 1m Q = 10-5 x 3 m3/sec/m (10b) For 5 such flow tubes Q = 5 x 10-5 x 3 m3/sec/m (10c) For a 25m wide dam Q = 25 x 5 x 10-5 x 3 m3/sec (10d) Calculation of flow Phreatic line 15 m h = 15m h =12m h = 9m h = 6m h = 3m h = 0 Note that per metre width (10e)

Slide 43: 

P 5m (11a) Calculation of pore pressure Phreatic line P 5m Pore pressure from 15 m h = 15m h = 12m h = 9m h = 6m h = 3m h = 0

Slide 44: 

P 5m (11a) (11b) Calculation of pore pressure Phreatic line P 5m Pore pressure from At P, using dam base as datum 15 m h = 15m h = 12m h = 9m h = 6m h = 3m h = 0

Slide 45: 

Example Calculating Pore Pressures 20 m 10 m

Slide 47: 

Step 1: Choose a convenient datum. In this example the sea floor has been chosen Then H1 = 40 m H2 = 1 m. The increment of head, h = 39/9 = 4.333 m

Slide 48: 

A B C D E Step 2: Calculate the head at points along the base of the vessel. For convenience these are chosen to be where the EPs meet the vessel (B to E) and at the vessel centerline (A). Hence calculate the pore water pressures. At B Head = H1 - 5 h = H2 + 4 h = 18.33 m Pore pressure at B = 18.33 w = 179.8 kPa

Slide 49: 

Step 3: Calculate the upthrust (Force/m) due to pore pressures = 3218 kN/m Without pumping Upthrust = 20  1  9.81 = 196 kN/m Upthrust due to Pumping = 3218 – 196 = 3022 kN/m

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