Laplace’s equation:
what does q out equal in 3D?
What about at steady state?
Qout = Qin + (dVx/dx + dVy/dy +dVz/dz)dxdy*dz
Qout = Qin
So, dVx/dx + dVy/dy +dVz/dz = 0
Darcy’s law (3D);
Vx, Vy, Vz =?
general equation?
what if K is equal in all direction?
Vx = Kx*ix i= hydrulic gradient Vy = Ky*iy. k= permeability Vz = Kz*iz. v=flow velocity
Kx(d2h/dx2) + Ky(d2h/dy2) + Kz*(d2h/dz2)
d2h/dx2 + d2h/dy2 + d2h/dz2 = 0
What is planar flow?
What causes this?
when construction is long compared to its width you can assume the flow lengthways is zero. So flow travels in one direction and is planer.
Cuttings
Embankments
Trenches
What is a flownet?
Flow nets are made up of a network of intersection lines representing “flowlines” and “equipotentials”. These intersect at right angles.
Flowlines: the trajectory of imaginary water
Equipotentials: lines which join points that have the same head.
Usings flow nets how do you calculate total flow (qt)?
qt = k*H* (Nf/Nh). (units M3/s per meter) Nf = flow tubes (spaces between flowlines) Nh = Equiptoential drops (spaces between equipotentials) H= total head drop.
State the criteria for flownets to satisfy Laplace’s equation.
- Boundary conditions must be satisfied.
- Flowlines must intersect equipotentials at right angles
- The cells must be ‘curvilinear’ (fit a circle in and it touches all sides.
- The quantity of flow through each flow channel is constant
- the headloss between each consecutive equipotential is constant
- flowlines can’t intersect with other flowlines
- An equipotential can intersect another equipotential
What is a confined flownet?
a flownet that all of its boundaries are defined and fixed at the outset. E.G. flownet of flow under a sheet pile wall with horizontal GWLs each side.
how to construct a flownet?
- Draw the structure and soil mass to a suitable scale.
- identify permeable and impermeable boundaries
- Use symmetry where it exists, to reduce the extent of flownet required to be drawn
- using a pencil, sketch a few flow lines in there ENTIRETY. the sketch appropriate number of equipotentials to form curvilinear squares.
- sketch the entire flownet before making adjustments
- you can add more flowlines and equipotential to give greater detail.
How can you find the pore pressure (u) using a flownet?
u = (h - z) * yw
yw = unit weight of water
h=head at a point
z= distance of the point from datum
Examples of unconfined flownets:
What is a phreatic surface (GWL)?
- the flow of groundwater to a river
- leakage from a canal
- seepage through an earth dam
Phreatic sources are the air/water boundary and are also the top flowline.
Tips to tackle unconfined flownets:
- Casagrande said top flowlines can be approximated as a parabola. (don’t get hung up on it.)
- Draw top flowline
- Mark Equipotnetial intersections
- Sketch flowlines and equipotentials
- Adjust to obtain curvilinear square
- Label equipotentials with their heads
what is the Laplace’s (2D) equation for anisotropic soil?
d2h/dx’2 + dh2/dZ2 = 0
Where x’ = x * sqrt( Kz/Kx )
What are seepage forces?
String upward seepage of water can cause pore pressures to be greater than gravity stress due to soil weight. This can cause the soils to fluidized or boil leading to instability.
How to calculate the critical hydraulic gradient?
icrit = hcrit / Z = (Y - Yw) / Y Y = unit weight
How to calculate the factor of safety in seepage induced instability of infinite slopes?
F = tan(phy') / tan(B) * ( 1 - u / (Z * Y * cos^2(B)) ) phy = mobilized angle B = slope angle Z = slope depth Y= soil unit weight or F = tan(phy') / tan(B) * ( 1 - (Yw*h) / (Z * Y * cos^2(B)) ) h = hieght of GWL
what is the factor of safety when the soil is drained?
F = tan(phy’) / tan(B)
