CE 162- Confined Flow Problems 1

7/27/2019 CE 162- Confined Flow Problems 1

http://slidepdf.com/reader/full/ce-162-confined-flow-problems-1 1/28

SEEPAGE PROBLEMS

PART 1: CONFINED FLOW PROBLEMS

CE 162

1st Semester 2013-2014



OVERVIEW

• Continuity Equation & Laplace’s Equation

• Flow Nets

•

Seepage calculations from flow nets• Flows nets in anisotropic soils

• Uplift Pressures

• Erosion and Piping



CONTINUITY EQUATION

CE 216Geotechnical Engineering I

1st Sem. 2013-2014



Continuity Equation



Continuity Equation

v = ki = k Ñh = k ¶h

¶ xe x +

¶h

¶ ye y

æ

è ç ö

ø÷

D q =Ñ×v = k ¶2h

¶ x2+

¶2h

¶ y2

æ

è ç ö

ø÷= 0

Ñ2h = 0 Ñ =¶

¶ xe x +

¶

¶ ye y

æ

è ç ö

ø÷



FLOW NETS IN ISOTROPIC SOILS


1st Sem. 2013-2014



• Boundary Types (Drawn in ink)

– Head (Dirichlet/Essential)

– Flow (Neuman/Natural)

• Flow Net lines (Drawn in pencil) – Equipotential lines

– Flow lines

–Equipotential lines perpendicular to flow lines

• Elements/Cells : region bounded by 2 parallelflow lines and 2 parallel equipotential lines.

Flow Nets



Flow Net



• Limit number of flow channels to 3.

• Rotate problem to check compliance with

square and perpendicular conditions.

• Limit partial drops to 1.

• Subdivide elements in half

–

Infinite element (2 sides) – 3-sided irregular element

– 5-sided irregular element

Flow Net Techniques



• Draw problem boundaries to scale in ink.

• Draw flow lines.

•

Draw equipotential lines• Repeatedly adjust flow lines and equipotential

lines until flow net are roughly perpendicular

to each other and elements are approximately

square.

Constructing flow nets



BASIC FLOW NET COMPUTATIONS


1st Sem. 2013-2014



Rate of Seepage

q = k D H n f

nd

where

q= flow rate per unit width

k =

permeabilityD H = head drop per equipotential line

n f = number of flow channels

nd = number of drops



• Given a point within the flow net, extrapolate

the head and elevation relative to the datum.

• Note prescribed heads and elevation should

be measured to the datum.

• Use definition of hydraulic head to compute

the pore pressure

Computing pore pressures

u = g w h - z ( )



• For a given point within the flow net, use thesubdivision technique to create an reducedelement in which the point is roughly in the

center.• Compute the head drop within the subdivided

element.

• Estimate with length of distance along the

direction of flow in the reduced element.• Divide the head drop by the flow distance to

estimate the hydraulic gradient.

Computing hydraulic gradients



FLOW NETS IN ANISOTROPIC SOILS


1st Sem. 2013-2014



Flow nets in anisotropic soils

Ñ×k ×Ñh = 0¶

¶ xk xx

¶h

¶ x+ k

xy

¶h

¶ y

æ

è ç

ö

ø÷

+¶

¶ yk yx

¶h

¶ x+ k

yy

¶h

¶ y

æ

è ç

ö

ø÷

= 0

k xx¶2h

¶ x2+ k yy

¶2h

¶ y2= 0

k xx

k yy

¶2h

¶ x2+

¶2h

¶ y2= 0

¶2h

¶ x 2 +

¶2h

¶ y2 = 0 x =k yy

k xx x = k x

¶

¶ x =¶

¶ x

¶ x

¶ x = k¶

¶ x

¶2

¶ x2 = k 2 ¶2

¶ x 2




• Scale in the direction of minimum permeabilityneeds to be stretched by the scale factor.

• Computations involving rate of seepage can bedirectly made from the flow net in the “stretchedscale”.

•

Flow net must be redrawn into equal scale inorder to compute pore pressures and hydraulicgradients.

q = k min

k max

D H n f

nd



UPLIFT PRESSURE ON STRUCTURES

CE 216

Geotechnical Engineering I

1st Sem. 2013-2014



SEEPAGE FORCES

CE 216

Geotechnical Engineering I

1st Sem. 2013-2014



EROSION AND PIPING

CE 266

Earth Structures

1st Sem. 2012-2013



THE END

THANK YOU

CE 162

1st

Semester 2013-2014

CE 162- Confined Flow Problems 1

Documents

Transcript of CE 162- Confined Flow Problems 1