Overview
Darcy's law is to groundwater what Ohm's law is to circuits: a simple proportionality between a flow and the push that drives it. Almost every calculation in the rest of the course, from flow to a well to the spread of a contaminant plume, starts here.
- Compute hydraulic head from elevation and pressure, and use it to decide the direction of flow.
- Apply Darcy's law to find discharge, Darcy flux and average pore-water velocity, and estimate travel times.
- Distinguish hydraulic conductivity from intrinsic permeability and convert between them for different fluids.
- Write Darcy's law in three dimensions for an anisotropic medium.
- Compute the equivalent horizontal and vertical conductivity of a layered system, and the refraction of flow lines at a layer boundary.
- Estimate K from grain size (Hazen, Kozeny–Carman), from constant- and falling-head permeameter tests, and for fractured rock (cubic law).
- Check whether Darcy's law applies, using the REV concept and the Reynolds number.
How to work through it: as in Module 1, each section has an animation, a reading and an interactive tool. Videos tagged required are part of the module; optional ones go deeper. Then do the practice problems, the Brightspace quiz, the OpenHydroQual exercise, and Assignments 3 and 4.
1Darcy's law and hydraulic head
Darcy's experiment
In 1856 the French engineer Henry Darcy measured the flow of water through vertical columns of sand while designing filters for the public fountains of Dijon. He found that the discharge \(Q\) is proportional to the cross-sectional area \(A\) of the column and to the difference in head \(\Delta h\) between its ends, and inversely proportional to its length \(L\). With a constant of proportionality \(K\), the hydraulic conductivity:
\(dh/dl\) is the hydraulic gradient (dimensionless). The minus sign says flow goes in the direction of decreasing head. \(K\) has units of velocity (m/day, cm/s, ft/day).
Hydraulic head
The head at a point is the elevation to which water rises in a piezometer open at that point. It is the mechanical energy per unit weight of water:
\(z\) is the elevation above a datum and \(p/\rho g\) the pressure head. Water flows from high head to low head, not necessarily from high pressure to low pressure, nor downhill. In a discharge area groundwater flows upward; under a hill it can flow toward points of higher pressure.
optional Tom Gleeson: hydraulic gradient in nested piezometers (12 min), the vertical direction of flow from piezometers at different depths.
Darcy flux and pore-water velocity
Dividing by the area gives the specific discharge or Darcy flux, the flow per unit of total cross-section:
Water can only pass through the connected pores, a fraction \(n_e\) (the effective porosity) of the cross-section, so the average pore-water (seepage) velocity \(v\) is larger than \(q\). Use \(q\) for discharges and water budgets; use \(v\) for travel times and for how fast a dissolved contaminant is carried.
Example 3.3. A culvert 5 m long and 0.8 m in diameter has been packed with sand (\(K = 3\) m/day, \(n_e = 0.38\)) by a storm. The culvert is submerged and the water level at one end is 1.6 m higher than at the other. Then \(A = \pi(0.4)^2 = 0.503\) m², \(dh/dl = 1.6/5 = 0.32\), \(Q = 3(0.503)(0.32) = 0.483\) m³/day, \(q = 0.96\) m/day and \(v = 0.96/0.38 = 2.53\) m/day. Water takes about two days to get through.
Virtual Darcy column
Total head, elevation head and pressure head along the column
Predict, then check: tilt the column so the outlet is higher. Does the flow reverse?
Predict, then check: when does the pressure head go negative?
Predict, then check: make K very large with coarse grains. What happens to the Reynolds number?
2Hydraulic conductivity, permeability and Darcy's law in 3-D
Intrinsic permeability
Hubbert (1956) noted that \(K\) depends on both the medium and the fluid. Flow is faster for a denser fluid (larger driving weight \(\rho g\)), slower for a more viscous one (\(\mu\)), and proportional to the square of the mean grain diameter \(d\):
\(k\) is the intrinsic permeability, a property of the porous medium alone, with units of area (m², cm²). Petroleum engineers use the darcy: 1 darcy \(= 9.87\times10^{-9}\) cm² \(= 9.87\times10^{-13}\) m². For water at 20 °C, \(\rho g/\mu \approx 9.8\times10^{6}\) m⁻¹s⁻¹, so 1 darcy corresponds to \(K \approx 10^{-5}\) m/s \(\approx 0.84\) m/day.
Example 3.4. How much more conductive is a sand to PCE (tetrachloroethylene: \(\rho = 1.63\) g/cm³, \(\mu = 9.0\times10^{-4}\) N·s/m²) than to water (\(\rho = 1.0\) g/cm³, \(\mu = 1.0\times10^{-3}\) N·s/m²)? With the same \(k\), \(\dfrac{K_{PCE}}{K_w} = \dfrac{\rho_{PCE}}{\rho_w}\dfrac{\mu_w}{\mu_{PCE}} = 1.63 \times \dfrac{1.0}{0.9} = 1.81\). Dense, low-viscosity solvents move through aquifers faster than water under the same gradient.
Typical values
\(K\) varies over more orders of magnitude than almost any other physical property used in engineering (Fetter, 2001):
| Material | K (cm/s) | K (m/day) |
|---|---|---|
| Clay | 10⁻⁹ – 10⁻⁶ | 10⁻⁶ – 10⁻³ |
| Silt, sandy silt, clayey sand, till | 10⁻⁶ – 10⁻⁴ | 10⁻³ – 0.1 |
| Silty sand, fine sand | 10⁻⁵ – 10⁻³ | 0.01 – 1 |
| Well-sorted sand, glacial outwash | 10⁻³ – 10⁻¹ | 1 – 100 |
| Well-sorted gravel | 10⁻² – 1 | 10 – 1000 |
Homogeneity, isotropy and Darcy's law in three dimensions
A medium is homogeneous if \(K\) is the same at every point, otherwise heterogeneous. It is isotropic if \(K\) at a point is the same in every direction, otherwise anisotropic. Flow in an aquifer is three-dimensional. When the coordinate axes are aligned with the principal directions of \(K\):
In vector form \(\mathbf{q} = -\mathbf{K}\,\nabla h\) with \(\mathbf{K} = \mathrm{diag}(K_x, K_y, K_z)\), and the magnitude of the flux is \(|\mathbf{q}| = \sqrt{q_x^2+q_y^2+q_z^2}\). This is where the gradient from Module 2 comes in: in an isotropic medium \(\mathbf{q}\) points straight down the head gradient, perpendicular to the contours of equal head. In an anisotropic medium it is turned toward the direction of larger \(K\).
Example 3.5 (simplified). In an unconfined sand, the horizontal head gradient is \(\partial h/\partial x = -0.0011\) and \(K_x = 8\) ft/day, so \(q_x = 0.0088\) ft/day. Recharge of 1.25 ft/yr \(= 0.00342\) ft/day crosses the water table downward. If piezometers show \(\partial h/\partial z = 0.03\) (head increasing upward, flow downward), then \(K_z = 0.00342/0.03 = 0.114\) ft/day: about 70 times smaller than \(K_x\). The total flux is \(\sqrt{0.0088^2 + 0.00342^2} = 0.0094\) ft/day, dipping \(\tan^{-1}(0.00342/0.0088) = 21°\) below horizontal. You will do the full three-piezometer version in Assignment 3.
3Layered systems and the refraction of flow lines
Equivalent hydraulic conductivity
Layering is the most common heterogeneity, because sediments are deposited in successive beds. We replace the stack of layers (thicknesses \(b_i\), conductivities \(K_i\)) by one equivalent, homogeneous but anisotropic medium.
Flow parallel to the layers. Every layer has the same gradient \(dh/dl\), so \(Q_i = K_i b_i\,(dh/dl)\) per unit width, and the flows add: \(\sum K_i b_i\,(dh/dl) = K_h\left(\sum b_i\right)(dh/dl)\).
Flow across the layers. The same flux \(q\) passes through every layer, so the head drops add: \(\Delta h = \sum \Delta h_i = q\sum b_i/K_i = q \sum b_i / K_v\).
The most permeable layer controls horizontal flow; the least permeable controls vertical flow. A single thin clay layer can make a thick sand nearly impermeable to vertical flow.
Layered-system calculator
Edit the layers, top to bottom. K in m/day, b in m.
Left: horizontal flux in each layer (same gradient). Right: head profile for vertical flow (same q).
Refraction of flow lines
Where flow crosses a boundary from \(K_1\) to \(K_2\), two conditions hold. The flux normal to the boundary is continuous (mass conservation), \(K_1\,\partial h/\partial n|_1 = K_2\,\partial h/\partial n|_2\). And the head is continuous along the boundary, so its tangential derivative is the same on both sides. Dividing one by the other, with angles measured from the normal:
Example 3.7. With \(K_1 = 10^{-6}\) cm/s, \(K_2 = 10^{-3}\) cm/s and \(\theta_1 = 5°\): \(\tan\theta_2 = \tan 5° \times 1000 = 87.5\), so \(\theta_2 = 89.3°\). Flow entering the aquifer turns almost parallel to it. This is why flow is nearly horizontal in aquifers and nearly vertical in aquitards.
A flow line crossing the boundary (K in m/day)
4Estimating and measuring K
optional Constant head permeability test, procedure and data analysis (7 min), same channel.
Answering almost any quantitative groundwater question requires \(K\). There are three ways to get it: empirical formulas from grain size, laboratory tests on samples, and field tests (pumping and slug tests, covered with well hydraulics). Field tests sample the largest volume and are usually the most representative.
Grain-size formulas
In Hazen's formula \(K\) is in cm/s and \(d_{10}\) in cm, with \(C\) between about 40 (very fine, poorly sorted sand) and 150 (coarse, well-sorted sand). It is meant for sands with \(d_{10}\) between about 0.1 and 3 mm. Kozeny–Carman uses a representative grain size \(d_m\) (often \(d_{50}\)) and the porosity, in any consistent units.
Example 3.8. A sand has \(d_{10} = 0.1\) mm, \(d_{50} = 0.24\) mm and \(n = 0.38\). Hazen with \(C = 100\)–\(150\): \(K = C(0.01)^2 = 0.010\)–\(0.015\) cm/s. Kozeny–Carman with \(d_{50}\): \(k = \dfrac{(2.4\times10^{-4})^2}{180}\,\dfrac{0.38^3}{0.62^2} = 4.57\times10^{-11}\) m², \(K = 9.81\times10^{6}\,k = 4.5\times10^{-4}\) m/s \(= 0.045\) cm/s. A factor of three or four between formulas is typical.
Permeameters
Constant head (sands, gravels): keep the head difference \(h\) across a sample of length \(L\) and area \(A\) fixed, and collect a volume \(V\) in time \(t\). Darcy's law gives directly
Falling head (silts, clays, where flow is too slow to collect): the head in a narrow tube of diameter \(d_t\) above a sample of diameter \(d_c\) falls as water drains through. The water leaving the tube equals the flow through the sample, \(-\frac{\pi d_t^2}{4}\frac{dh}{dt} = K\,\frac{\pi d_c^2}{4}\,\frac{h}{L}\). Separating variables and integrating from \(h_0\) at \(t = 0\):
Example 3.9. A sandy-silt sample 30 cm long and 10 cm in diameter, with an expected \(K \approx 10^{-5}\) cm/s, is tested with an 8 mm burette. The time for the head to halve is \(t = \dfrac{d_t^2 L}{d_c^2 K}\ln 2 = \dfrac{(0.8)^2(30)}{(10)^2(10^{-5})}(0.693) \approx 13{,}000\) s \(\approx 3.7\) hours.
Virtual falling-head permeameter
Each sample has an unknown K. Run the test, read the head at a few times, then compute K from the data, either from two readings or from the slope of ln(h₀/h) against t.
Your readings
| t (s) | h (cm) | ln(h₀/h) |
|---|
Flow in fractured rock: the cubic law
Seepage under dams, into tunnels and through crystalline rock happens mostly in fractures. If the fractures are idealized as smooth parallel plates with aperture \(b\), laminar flow between them gives, per fracture of width \(w\),
where \(N\) is the number of fractures per unit distance normal to them. Doubling the aperture multiplies the flow by eight. Real fractures are rough and variable, so the cubic law gives an order of magnitude.
Example 3.10. A granite has one fracture every 2 m (\(N = 0.5\) m⁻¹), aperture 0.3 mm: \(K = \dfrac{0.5(1000)(9.81)(3\times10^{-4})^3}{12(1.0\times10^{-3})} = 1.10\times10^{-5}\) m/s, about 1 m/day: the same as a fine sand, from rock that is almost entirely solid.
When does Darcy's law apply?
Scale. The representative elementary volume (REV) is the smallest volume over which a measured property is representative of the whole. Below the REV, porosity jumps between 0 (a grain) and 1 (a pore), and \(K\) is undefined. Darcy's law applies at the REV scale and above.
Laminar flow. Darcy's law assumes viscous forces dominate inertia. With the Reynolds number based on the Darcy flux and the mean grain diameter,
That covers nearly all natural groundwater flow. Exceptions are karst conduits, very coarse gravels, and the immediate vicinity of large pumping wells.
5Practice (ungraded)
New numbers every time. Aim for two correct attempts in a row on each type before taking the quiz. Scientific notation can be typed as 2.5e-4.
A · Darcy's law: discharge, flux, velocity, travel time
B · Hydraulic head and the direction of flow
C · Intrinsic permeability and other fluids
D · Darcy's law in 2-D with anisotropy
E · Equivalent K of a layered system
F · Refraction at a layer boundary
G · Permeameter tests
H · Hazen and Kozeny–Carman
I · Fractured rock (cubic law) and the Reynolds number
Concept checks
1. Piezometer A is screened 10 m deep with a water level 2.0 m below the ground; piezometer B, beside it, is screened 30 m deep with a water level 1.5 m below the ground. Which way is the vertical flow?
2. A sand aquifer 20 m thick contains a continuous clay layer 0.5 m thick. Which statement is correct?
3. A dye tracer travels 30 m between two wells in 10 days. A student uses 3 m/day as the Darcy flux \(q\) to compute the discharge through the aquifer. What is wrong?
6Graded quiz
When you can do every practice problem without opening the solution, go to Brightspace → Quizzes → Module 3: Darcy's Law and Hydraulic Conductivity.
- Head and flow direction, Darcy's law, permeability, layered systems, refraction and permeameters; numerical questions use different numbers for every student and attempt.
- 2 attempts, highest score counts. Calculator and notes allowed.
Then complete Assignment 3 (Darcy's law and permeability) and Assignment 4 (hydraulic conductivity and layered systems), linked on the course page, and submit your Excel workbooks in Brightspace.
7OpenHydroQual exercise: Darcy's column in a computer
You will run a numerical version of Darcy's experiment, a sand column split into ten cells between two reservoirs, check that the model reproduces Darcy's law, and then put a less permeable layer in the column and compare the model with the series formula for \(K_v\).