ATM 301 Lecture #7 (sections 7.3-7.4)

Soil Water Movements – Darcy’s Law and Richards Equation

infiltration (~76% of land precip.)

redistributionunsaturated soil

saturated 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)

Hydrologic Soil Horizons (or layers):

Phreatic Zone (saturated zone)

Vadose Zone(unsaturated zone)

p=w g (z’-z’o)

p=w g (z’-z’o)

Darcy’s Law: for saturated subsurface flowSpecific discharge is proportional to the pressure gradient:

dx

dhK

A

Qq hx

x

xx

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

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.

Hydrostatic

Downward flow

• If there is no pressure gradient over a distance, no flow occurs (these are hydrostatic conditions);

• If there is a pressure gradient, flow will occur from high pressure towards low pressure (opposite the direction of increasing gradient – hence the negative sign in Darcy's law);

• The greater the pressure 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.

Simple applications of the Darcy’s Law:

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 Ax

qx

Qx

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 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:

whered = average grain diameter in mv = 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.

v

dqxRe

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 diameterC = 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.

v

gkK Ih

General Saturated-Flow Equation:

dt

hS

z

hK

zy

hK

yx

hK

x shzhyhx

volumecontrolheadindecreaseorincrease

volumecontrolleavingorenteringwaterofvolumeSs

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.

1. Steady-state flow,

2. Isotropic and homogeneous medium, Khx=Khy=Khz =Kh (diffusion eq.):

3. Steady-state, isotropic and homogeneous medium (Laplace equation):

Simplified Forms of the Saturated Flow Equation:

0

z

hK

zy

hK

yx

hK

x hzhyhx

:0/ th

t

h

K

S

z

h

y

h

x

h

h

s

2

2

2

2

2

2

02

2

2

2

2

2

z

h

y

h

x

h

Hydrologic Soil Horizons (or layers):

Phreatic Zone (saturated zone)

Vadose Zone(unsaturated zone)

p=w g (z’-z’o)

p=w g (z’-z’o)

Darcy’s Law: for unsaturated subsurface flow

dx

zdKq hxx

)]([)(

where

q= local volumetric water content

Both Khx and pressure head (psi) increase with

The Kh vs. and vs. relationships are crucial

For the vertical z direction:

dz

dKq hzz

)(1)(

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

Fig. 1 Jet Fill Tensiometer from Soil Moisture Equipment Corp. (www.soilmoisture.com)

Moisture-characteristic curve

Moisture-conductivity curve

Approximated as:

()=ae (θ/)b

f= porosity or void fraction in a medium.

b = parameter that depends on pore size distribution

Kh = saturated hydraulic conductivity

h

b

h KK32

)(

Effect of water content () on pressure (tension head) and conductivityBoth Khx and pressure head increase with Tension head decreases with

Effect of water content on pressure (tension head) for different soils

• tension head decreases with grain size and with water content

Effect of water content on hydraulic conductivity for different soils

Maidement (1993)

Kh

Note:• Higher conductivity at higher water contents • Higher conductivities for coarse textures

qz=-Kh[1+d()/dz]

Saturated value =ae (θ/ϕ)b

General Unsaturated-Flow Equation: Richards Equation

tzK

zyK

yxK

x hhh

1

)()(

)()(

)()(

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:

hSs

xK

xx

zKq hhx

)(

)()(

)(

z

Kq hz

)(1)(

yK

yy

zKq hhy

)(

)()(

)(

Derive this Eq. from mass conservation andthe Darcy’s law following Box 7.3 on p. 326

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.

tzK

zz

Kh

h

)()(

)(

tzK

zz

Kh

h

'

)()(

''

)(

American Lorenzo Adolph Richards (1904–1993) was one of the 20th century’s most influential minds in the field of soil physics. He derivedthe Richard Eq. in 1931.

Home work #3 (on soil moisture; due Oct. 6):

1. Ex. 1 on p. 342 of Dingman (2015) (see section 7.1.1.1, Box 7.1, use spreadsheet) (25%)

2. Ex. 2 on p. 342 (see Section 7.1.2-7.1.4, Box 7.1) (15%)

3. Ex. 5 (Experiment A only) on p. 387 (see Section 8..4.3, Need spreadsheet, use text-disk program) (30%)

4. Briefly describe the Darcy’s law using plain language and an equation (15%)

5. Briefly describe the Richards equation using plain language and an equation (for vertical direction only, i.e., no horizontal flow case). (15%)