CIVL2410 Chap.7 Flow nets: Laplace's equation and curvilinear squares
Flow nets: Laplace's equation and curvilinear squares
Real seepage spreads in two dimensions: under a sheet pile, around a cut off wall, into an excavation from every side. The governing equation is Laplace's, and although it can be solved numerically it has a graphical solution that is quicker, more transparent and still examined, because sketching it forces the boundary conditions to be stated.
This chapter derives the equation, explains why its two families of curves must cross at right angles, and reduces the whole problem to two counts.
What this chapter covers
- 01
Darcy in two directions, and continuity as the only physics added
- 02
Laplace's equation, and why the permeability cancels out of it
- 03
What Laplace quietly assumes, and which assumption the consolidation chapter releases
- 04
The potential function, and equipotentials as lines of equal total head
- 05
The stream function, and flow lines as the paths water follows
- 06
Orthogonality, proved from the product of the two slopes
- 07
Curvilinear squares and the inscribed circle test
- 08
Why every flow channel carries the same discharge
- 09
Counting channels and drops into the discharge relation
- 10
The five sketching rules, and where a net either converges or goes wrong
Discharge under a cut off wall
- +1Convert line counts into the counts the formula wants. Five flow lines bound four channels, so Nf = 4.
- +1Thirteen equipotentials give twelve head drops, so Nh = 12.
- +1Q = k H Nf / Nh = 2.8 x 10^-5 x 5.4 x (4 / 12) = 5.04 x 10^-5 m3/s per metre run.
- +1Over a day that is 5.04 x 10^-5 x 86400 = 4.35 m3, about 4,350 litres per metre of wall.
Key terms
- Flow line
- A path a water particle follows through the soil. The space between two adjacent flow lines is a flow channel, and every channel in a properly drawn net carries the same discharge.
- Equipotential
- A line joining points of equal total head. The head drop between two adjacent equipotentials is the same everywhere in the net, which is what makes counting them meaningful.
- Curvilinear square
- A cell of the net whose average width equals its average length, tested by whether a circle can be inscribed touching all four sides. It is what makes the flow per channel independent of cell size.
- Laplace's equation
- The governing equation for steady seepage through a homogeneous isotropic soil, obtained from Darcy's law plus continuity. The same equation governs heat conduction, electrical current and diffusion.
Flow nets: Laplace's equation and curvilinear squares FAQ
Why does the permeability disappear from the equation?
Because for an isotropic soil it factors out of both terms and cancels. The consequence is worth stating explicitly in an answer: the shape of a flow net depends only on the geometry and the boundary conditions, never on the soil, so the same net serves a fine sand and a coarse gravel.
The permeability re enters only at the last step, when the counted net is converted into an actual discharge, which is also why an anisotropic soil needs the section redrawn before an ordinary net can be sketched on it.
Does drawing a finer net give a more accurate seepage?
No, and expecting it to is a sign that the counts have been misread. Doubling both the number of flow lines and the number of equipotentials leaves their ratio unchanged, so the computed discharge is identical.
A finer net improves the local resolution, which matters when you need the gradient at a particular exit point, but the total flow is a property of the geometry and the head difference and the mesh is only the instrument used to read it.
Exam move
Learn the derivation as three steps rather than as a result, because a question that asks where Laplace's equation comes from is asking for continuity plus Darcy and nothing else. When sketching, spend the first minute on boundary conditions and label them on the drawing: two of the four curves in most problems are boundaries and are therefore already drawn.
Enforce right angles at those boundaries before worrying about whether the cells look square, since an oblique intersection means the net is not a solution however even the mesh appears. Practise counting out loud, spaces rather than lines, until it is automatic, because a correct sketch feeding a wrong count is the commonest way this topic is lost.
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