CIVL2410 Chap.5 Head, hydraulic gradient and Darcy's law
Head, hydraulic gradient and Darcy's law
Every soil is permeable, because the grains touch rather than fuse and the void space is connected. That matters far beyond drainage design: pore water pressure controls effective stress, and effective stress controls strength, so no soil can be analysed without knowing what the water inside it is doing.
This chapter establishes the quantity that drives seepage, the gradient it produces, and the law that turns a gradient into a flow.
What this chapter covers
- 01
The water table, perched water and artesian conditions
- 02
Head as energy per unit weight, and what a piezometer physically measures
- 03
Bernoulli's three terms, and the arithmetic that removes the velocity term
- 04
Total head as pressure head plus elevation head, from a declared datum
- 05
Hydraulic gradient as head loss per unit length along the flow path
- 06
Darcy's law in its two forms, and the symbol list that goes with it
- 07
The coefficient of permeability as a property of the soil rather than the geometry
- 08
The four assumptions inside the law, and which one fails first
- 09
Intrinsic permeability, and why two symbols exist
- 10
Discharge velocity against seepage velocity, and which one a travel time needs
Flow through a sand layer between two reservoirs
- +1The two reservoir levels are the total heads at the ends of the flow path, so the head loss is 4.6 - 2.8 = 1.8 m.
- +1The gradient is that loss divided by the path length: i = 1.8 / 25 = 0.072.
- +1The gross area perpendicular to flow is A = 1.2 x 8 = 9.6 square metres.
- +1Darcy gives q = k i A = 3.5 x 10^-4 x 0.072 x 9.6 = 2.42 x 10^-4 m3/s, about 0.24 litres per second.
- +1For travel time use the seepage velocity: v = k i = 2.52 x 10^-5 m/s, so vs = v / n = 6.63 x 10^-5 m/s, and 25 m takes about 3.8 x 10^5 s, roughly 4.4 days.
Key terms
- Total head
- Pressure head plus elevation head at a point, measured from a declared datum. Water flows from high total head to low total head, which is not the same as flowing from high pressure to low pressure.
- Piezometer
- A narrow tube, open at the top and sealed into the ground at its tip. The height water stands in it is the total head at that tip, which is why a pair of them settles the direction of flow between two points.
- Hydraulic gradient
- Head loss per unit length along the flow path. It is dimensionless, and it is the term Darcy's law is proportional to.
- Discharge velocity
- Flow rate divided by the total cross sectional area, grains included. It is a bookkeeping rate rather than the speed any water particle travels.
- Seepage velocity
- The actual average speed of water through the pore space, obtained by dividing the discharge velocity by the porosity. It is always the larger of the two and is what a travel time or a contaminant arrival needs.
Head, hydraulic gradient and Darcy's law FAQ
Why is the velocity term in Bernoulli's equation ignored?
Because it is many orders of magnitude smaller than the terms beside it. Seepage velocities in sand are around 10^-4 to 10^-6 m/s, and at the fast end of that range the velocity head works out at around 5 x 10^-10 metres, while the pressure and elevation terms are measured in metres.
Dropping a term that small is not an approximation worth defending against, and keeping it adds false precision to a calculation whose permeability is rarely known better than a factor of two.
Two piezometers show different pressures. Which way does water flow?
You cannot tell from the pressures alone, and trying to is one of the most common errors in this topic. Flow runs from high total head to low total head, and total head is pressure head plus elevation head. A point deep in the ground can have a much higher pore pressure than one near the surface while having a lower total head, because the elevation term more than makes up the difference.
Read the two standpipe levels, not the two pressures.
When does Darcy's law stop being valid?
It assumes laminar flow at low Reynolds number, a saturated soil with every pore filled, a homogeneous and isotropic material, and steady state conditions.
In practice the first is broken by coarse gravel under a steep gradient, where flow turns turbulent and discharge stops being proportional to gradient; the second by the unsaturated zone above a water table; the third by layering, which the anisotropy chapter deals with; and the fourth by a consolidating clay, which is what the consolidation chapter is about.
Exam move
Fix the definition of total head first and rehearse it until the datum discipline is automatic, because almost every error in this module is a depth or an elevation measured from the wrong reference rather than a misunderstanding of the law. Sketch the section for every problem and write the total head at each point where you know it before touching Darcy's law.
Learn the two velocities as answers to two different questions, quantity of water against travel time, and decide which one the question wants before substituting. Finally, memorise the permeability ladder by soil type well enough to sanity check any computed value, since a permeability outside that range is an arithmetic slip rather than an unusual soil.
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