1. Darcy’s Law and Richards Equation
ATM 301 Lecture #7 (sections )
Soil Water Movements –
Darcy’s Law and Richards Equation

2. infiltration (~76% of land precip.)
redistribution
unsaturated soil
saturated zone with groundwater
Overland flow
The law governing water flow through a porous medium such as soil was discovered in 1856 by Henri Darcy, a French engineer.
Henri Darcy
(6/10/1803–1/03/1858)

3. Hydrologic Soil Horizons (or layers):
Vadose Zone
(unsaturated zone)
p=w g (z’-z’o)
Phreatic Zone
(saturated zone)
p=w g (z’-z’o)

4. Darcy’s Law: for saturated subsurface flow
Specific discharge is proportional to the pressure gradient:
where
qx = specific discharge (discharge per unit area, in m/s), also called Darcy velocity
Qx = volume discharge in m3/s
Khx = saturated hydraulic conductivity in m/s
dh/dx = the gradient of total hydraulic head, h

5. Hydraulic Head (h) in ground and soil water:
Hydraulic head (or simply head), h, is the fluid potential, or the mechanic energy per unit weight of the fluid (unit: in meters)
h = z +  = z + p/ = z + p/( g)
z = elevation (m, gravitational head)
p = total pressure (including air pressure, N/m2)
 (psi)= p/( g) is the pressure head (in m)
The total head h = z +  in a saturated flow can be measured as the height to which water rises in a piezometer, a tube connecting the point to the air.
For the vertical direction, Darcy’s law becomes:

6. Porosity and actual flow velocity:
Porosity (, phi) is the ratio of the void space (filled with air or water) between grains or particles within a porous medium divided by the volume of the medium.
 ranges from ~40% for sandy soil with large particles to close to 50% for clay soil with fine particles.
Flow velocity (Ux, m/s) of a fluid within a medium is related to the Darcy velocity qx
Ux  Qx /( Ax) = qx /  Ux qx Ax Qx

7. Simple applications of the Darcy’s Law:
Hydrostatic
Downward flow
If there is no gradient in hydraulic head (h) over a distance,
no flow occurs (these are hydrostatic conditions);
If there is a gradient in h, flow will occur from high head
towards low head (opposite the direction of
increasing gradient –> the negative sign in Darcy's law);
The greater the head gradient (through the same
formation material), the greater the discharge rate; and
The discharge rate of fluid will often be different —
through different formation materials (or even through
the same material, in a different direction) — even if the
same pressure gradient exists in both cases.

8. Darcy’s Law: Limitations
The specific discharge or Darcy velocity (qx ) represents the averaged flow speed over a small local volume of the order of, or larger than, several grain/particle diameters (representative elementary volume or REV).
Darcy’s law does not hold on scales smaller than this REV scale.
Darcy’s law only applies to parallel or laminar flows.
As the flow velocity increases, nonlinear relation arises between the flow rate and head gradient. This is usually quantified using the Reynolds Number, Re:
where
d = average grain diameter in m
v = kinematic viscosity of the fluid (1.7x10-6m2/s)
v = /,  = (dynamic) viscosity discussed earlier.
Think of viscosity as “resistance to flow”!
Darcy’s law is valid when Re < 1, which is true for most flows in the ground and soil.

9. Permeability and Hydraulic Conductivity:
The hydraulic conductivity (Kh) depends on the nature of the medium (e.g., fine vs. coarse grain) and type of the fluid (e.g., water vs. oil):
and
kI = C d2 is the intrinsic permeability that depends only on the nature of the medium.
g = gravitational acceleration (9.8m/s2)
d = the average grain diameter
C = parameter that depends on grain shape, size distribution and packing.
v = kinematic viscosity (1.7x10-6m2/s)
See p. 325 on how Kh is measured.

10. General Saturated-Flow Equation:
where
Ss is the specific storage (in m):
Ss depends on the compressibility of the fluid and of the medium.
Derive it using mass conservation and the Darcy’s law (see Box.7.3 on pp ).

11. Simplified Forms of the Saturated Flow Equation:
Steady-state flow,
2. Isotropic and homogeneous medium, Khx=Khy=Khz =Kh (diffusion eq.):
3. Steady-state, isotropic and homogeneous medium (Laplace equation):

12. Hydrologic Soil Horizons (or layers):
Vadose Zone
(unsaturated zone)
p=w g (z’-z’o)
Phreatic Zone
(saturated zone)
p=w g (z’-z’o)

13. Darcy’s Law: for unsaturated subsurface flow
where
= local volumetric water content
Both hydraulic conductivity Khx and pressure head  (psi) increase with 
The Kh vs.  and  vs.  relationships are crucial
For the vertical z direction:

14. Soil-water Pressure Head  in unsaturated (or vadose) zone:
Capillary Rise (or Capillary Action)
is the ability of a liquid to flow in narrow spaces without the assistance of, and in opposition to, external forces like gravity. 
If the diameter of the narrow space (e.g., within a tube or between soil particles) is sufficiently small, then the combination of surface tension and adhesive forces between the liquid and container (or particles) act to lift the liquid. 
Inside the unsaturated soil layer, the capillary rise causes tension or negative (i.e., less than atmospheric, or inward) pressure on the water surface.
This pressure is measured using tensiometers
is called tension head. pm

15. Fig. 1 Jet Fill Tensiometer from Soil Moisture Equipment Corp. (www
Fig. 1 Jet Fill Tensiometer from Soil Moisture Equipment Corp. (

16. Effect of water content () on pressure (tension head) and conductivity
Both Khx and pressure head  increase with 
Tension head decreases with 
Approximated as:
()=ae (θ/)b
= porosity or void fraction in a medium.
b = parameter that depends on pore size distribution
Kh = saturated hydraulic conductivity
Moisture-characteristic curve
Moisture-conductivity curve

17. Effect of water content on pressure (tension head) for different soils
tension head decreases with grain size and with water content

18. Effect of water content on hydraulic conductivity for different soilsKh
Note:
Higher conductivity at higher water contents
Higher conductivities for coarse textures
Maidement (1993)

19. qz=-Kh[1+d()/dz]
Saturated value
Porosity
=ae (θ/)b

20. General Unsaturated-Flow Equation: Richards Equation
for isotropic Kh and the z direction is vertical.
The specific storage (in m):
In unsaturated soil, Ss depends on changes in water content relative to hydraulic head, not the compressibility of the fluid and of the medium.
And Darcy’s law becomes:
Derive this Eq. from mass conservation and
the Darcy’s law following Box 7.3 on p. 326

21. Solutions to the General Flow Equation:
Given the Kh() and () relations and boundary conditions,
the Richards equation can be solved using numerical methods for (x,y,z) and (x,y,z)= (), and the head as h(x,y,z)= (x,y,z) + z
Vertical downward flow (percolation): no horizontal flow, most common application (Also referred to as the Richards Eq.): or
where z’=-z pointing downward, i.e., increasing downward.
This is also solved numerically on computers.
American Lorenzo Adolph Richards (1904–1993) was one of the 20th century’s most influential minds in the field of soil physics. He derived
the Richard Eq. in 1931.

22. Home work #3 (on soil moisture; due on Oct. 3):
Ex. 1 on p. 342 of Dingman (2015) (see section , Box 7.1, use spreadsheet) (25%)
Ex. 2 on p (see Section , Box 7.1) (15%)
Ex. 5 (Experiment A only) on p. 387 (see Section , Need spreadsheet, use text-disk program) (30%)
Briefly describe the Darcy’s law using plain language and an equation (15%)
Briefly describe the Richards equation using plain language and an equation (for vertical direction only, i.e., no horizontal flow case). (15%)