Outline. Seepage Analysis. Soil Mechanics. Laplace Equation of Continuity
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1 Soil Mecanics Seepage nalysis Ci-Ping Lin National Ciao Tung Uni. Outline Laplace Equation of Continuity 1-D D Eample (-D) Computations using in nisotropic Soil Seepage troug an Eart Dam 3-D D Flow
![Continuity [ ] 0 d dy](/docs-images/78/77277159/images/2-1.jpg "ddy d dy d ddy d 0 i i")
2 Laplace s Equation of Continuity Laplace Eqn. Laplace s Equation of Continuity [ ] 0 d dy ddy d dy d ddy d 0 i i 0 0 If y Laplace Eqn.
3 1-D Eample 0 C C 1 t 0, 1, we can get C 1 1. ( ) C / C / t, 1 ) ( ( ) C / C / t, t, 0, and we get ( ) ( ) C C / 1 ) ( ) ( Laplace Eqn. t L, (L ) (L ), so we can get / / 1 1 ) ( 1 ( ) 1 For 0 < L, For L < L L, Laplace Eqn.
4 Solution to -D flow nalytical solution possible only for simple boundary conditions. ut most seepage problems ae comple boundary conditions. lternatie solution Flow net (Grapical solution) Electrical analogy models Numerical solutions Flow troug a Dam Preatic line Unsaturated Soil Drainage blanet Flow of water 0
5 Grapical representation of solution 1. Equipotentials Lines of constant ead, (,) Equipotential (EP) Grapical representation of solution. Flow lines Pats followed by water particles - tangential to flow Preatic line Flow line (FL) Equipotential (EP)
6 Properties of Equipotentials Flow line (FL) Equipotential (EP) Tus: Equipotenial slope (,) constant d d d d 0 EP / / (1a) (1b) (1c) Properties of Flow Lines Geometry Flow line (FL) Equipotential (EP) Kinematics From te geometry d d FL (b) Now from Darcy s law ence d d FL (c)
7 Ortogonality of flow and equipotential lines Flow line (FL) Equipotential (EP) On an equipotential d d EP / / On a flow line d d FL ence d d FL d d EP 1 (3) Geometric properties of flow nets Y Q y From te definition of flow (4a) t y X Z EP X Conclusion FL T Q FL From Darcy s law Q t Combining (4a)&(4b) y t (4b) (4c) Q y t YX (5) ZT Similarly Q YX ZT (4d)
8 Geometric properties of flow nets b Q c d a Q FL C Conclusion cd ab CD D EP( ) EP ( ) From te definition of flow Q cd From Darcy s law Q ab Combining (6a)&(6b) cd ab Similarly Q CD (6a) (6b) (6c) (6d) Grapical Construction of Flow Net oundary conditions: Submerged soil boundary Equipotential b. Impermeable soil boundary - Flow Line c. Line of constant pore pressure - eg. preatic surface 1. Equipotential line flow line. Flow element ~ square
9 Procedure for drawing flow nets Mar all boundary conditions Draw a coarse net wic is consistent wit te boundary conditions and wic as ortogonal equipotentials and flow lines. (It is usually easier to isualise te pattern of flow so start by drawing te flow lines). Modify te mes so tat it meets te conditions outlined aboe and so tat rectangles between adjacent flow lines and equipotentials are square. Refine te flow net by repeating te preious step.
10 Calculation of flow Computations using Preatic line 15 m 15m 0 1m 9m 6m 3m For a single Flow tube of widt 1m: For 10-5 m/s and a widt of 1m For 5 suc flow tubes For a 5m wide dam Note tat per metre widt Q Q m 3 /sec/m Q m 3 /sec/m Q m 3 /sec Q N N f Computations using Calculation of pore pressure Preatic line 15 m 5m 15m P 0 Pore pressure from t P, using dam base as datum 1m 9m 6m u w u w γ w [ 1 ( 5)] γ 3m w
11 Computations using 001 roos/cole, a diision of Tomson Learning, Inc. Tomson Learning is a trademar used erein under license. Figure 7.13 (a) weir; (b) uplift under a ydraulic structure nisotropic in nisotropic Soil Goerning Equation V 0 Transformation α 0 α V α V 0
12 nisotropic Eample1: Flow net for anisotropic soil Te figure sows te dam drawn at its natural scale Impermeable 1 dam L Soil layer Z Impermeable bedroc Eample1: Flow net for anisotropic soil Let us assume tat te soil as different oriontal and ertical permeabilities suc tat 4 V Transformation nisotropic α 4 V V V so or
13 nisotropic Eample1: Flow net for anisotropic soil Te figure sows te dam drawn to its transformed scale 1 L/ Soil layer Z Impermeable bedroc nisotropic Eample: Flow net for anisotropic soil
14 nisotropic Equialent permeability for anisotropic flow - - Q Natural scale t transformed scale Considering oriontal flow we ae (a) Natural scale Q t (b) Transformed scale Q eq t eq t V (7a) (7b) Equating 7a and 7b gies eq V nisotropic Eample1: Seepage under a dam m.5 m V 10-6 m/s m/s tus eq ( 4 10 ) ( 10 ) 10 m/sec ( 13. 5) 075. m Q ( 10 ) ( 0. 75) m / s / m Q m / s / m 9 10 m / s / m
15 Seepage in Eart Dam Seepage troug a dam on imperious base Seepage in Eart Dam 1. Obtain α. Calculate and ten Calculate d 4. Wit now alues of α and d, calculate L by 5. Wit nown alue of L, calculate q by
16 Seepage in Eart Dam Seepage troug a dam on imperious base 3-Dimentional Flow 3-D Flow Laplace Eqn.. becomes y 0 Difficult to sole for most 3-dimentional 3 problems. Eception: flow to wells Pumping tests-confined acquifer Pumping tests-unconfined aquifer
17 Pumping Tests-Confined quifer 3-D Flow 3-D Flow Pumping Tests-Unconfined quifer
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