Seepage Chapter 7 Laplaces Equation 1 2/28/2015 Laplaces Equation - PDF document

2/28/2015 Seepage Chapter 7 Laplace’s Equation 1

2/28/2015 Laplace’s Equation � � ���� A � ��� � � � ���� � ��� � Laplace’s Equation No storage No drainage  Darcy’s Law  Anisotropic Soil  Isotropic Soil  2

2/28/2015 Solution of Laplace’s Equation Since water can Flow Line only flow due to a change in head, and equipotential lines are lines of constant head, the flow lines must be perpendicular to the equipotential Equipotential Line lines everywhere. h = constant Flow Nets h = H 1 h = H 2 Upstream and downstream surfaces are equipotential lines 3

2/28/2015 Flow Nets q = 0 q = 0 All impervious boundaries are flow lines Flow Nets Flow lines and equipotential lines are orthogonal 4

2/28/2015 Flow Nets “Square” Ideally, the “cells” in the flow net should be “square” Flow Net Example Flow lines are  to upstream ground surface Flow lines are  to face of toe filter faces Equipotential lines are  to impervious boundaries Flow lines are  to equipotential lines everywhere “Cells in the flow net are “square” everywhere 5

2/28/2015 Common Flow Net Mistakes Flow Lines Not Perpendicular Extraneous Equipotential Lines Not Square Common Flow Net Mistakes Not a right angle Disappearing Flow Lines Equipotential Lines Not Perpendicular 6

2/28/2015 Correctly Drawn Flow Net Seepage Calculations ∆� � � �� � � � � � � � � � � � � � 1 � � � � � � � � � 7

2/28/2015 Seepage Calculations Seepage Calculations ∆� � � � � � � � � � � � � � � � � � � � � � � ⋯ 8

2/28/2015 Seepage Calculations ∆� � � � Total head loss � � # of head drops Seepage Calculations � ∆� � �� � 3 flow channels � � � � � � 8 head drops H ⑧ ❶ ❷ ❸ ① ② ③ ④ ⑤ ⑥ ⑦ 9

2/28/2015 Flow Nets Flow Nets 10

2/28/2015 Example 1. What is the head at points a, b, c, d and e ? 2. What is the seepage rate beneath the dam? 16 m e a c � � 3 � 10 �� ��/��� b d Partial Flow Channels 11

2/28/2015 Partial Flow Channels ∆� � � � � � � � � � � � � � � � � � � � � � � � � � ⋯ Partial Flow Channels 12

2/28/2015 Partial Flow Channels 2.38 flow channels 6 head drops � � � 1 2.38 flow channels � � � 1 � � � 0.38 13