WATER IN THE SOIL
 Soil is a porous media composed by solid, water, and air
SPATIAL CHARACTERIZATION
Vadose Zone ≈ 10 cm – 100 m
Capillary fringe ≈ 1 cm in Sand
10 m in Clay
Vadose
Zone
Root
Zone
Unsaturated
Zone
Capillary Fringe
Saturated
Zone
 “Groundwater hydrology”, is historically separated by surface hydrology
Unsaturated zone (≈ Vadose zone)
It represents the source of moisture for vegetation. In this zone evaporation,
transpiration and recharge to deeper saturated aquifers occur.
It controls the separation of precipitation between infiltration into soil and
surface runoff, therefore it is a “hub” for all of the hydrological processes.
It is a highly non-linear system with many components (water, vapor, air)
Saturated Zone
Aquifer  is an underground layer of water-bearing permeable rock or
unconsolidated materials from which groundwater can be extracted relatively
easily.
Aquiclude (aquifuge)  An impermeable body of rock or stratum of sediment
that acts as a barrier to the flow
Aquitard  A rather impermeable body of rock or stratum of sediment with
very low permeability.
Water is mainly stored in unconsolidated porous material:
 Alluvial, colluvial deposits (Typical aquifers)
 Sandstone, Limestone, Karst formation (Fractured aquifers)
 Volcanic rock, shale clay (Basically impermeable)
TYPICAL VALLEY SECTION
Spring
Perched
aquifer
30-50 m
Valley aquifer
Soil - Porous
material
Fractured
rock
Near surface  Soil ≈ 5 cm – 30 m; mainly composed of sand-silt-clay and a
fraction of organic material.
Greater depths  Aquifers; saturated soil defined as a “body” from where water
can be relatively fast and easily withdrawn.
A typical aquifer is in unconsolidated material (gravel, sand, fine-sand) of alluvial
or glacial origins. Fractured-rock and karst aquifers also exist and depend on the
degree of fracturing.
PROPERTIES OF POROUS MATERIALS
s: solids
v: voids
w: water
a: air
V  Va  Vw  Vs
V
SOIL POROSITY
Vv  V  Vs  Va  Vw
Representative Elementary Volume: A sufficiently large volume of soil
containing a large number of pores, such that the concept of mean global
properties is applicable but it is still small enough to be homogenous
 Vv 
n  lim 

V 0 V


Porosity
Porosity is a function of particle size and rearrangement of the particles (for
specific soil, such as clay or organic soil also of electrical charges)
Porosity can be artificially increased and decreased (raking, ploughing,
compaction, stock treading)
Water content
WATER CONTENT
(Volumetric)
 Vw 
  lim 

V 0 V


 sat
Saturated Water Content ≈ n, water content at full saturation
r
Residual Water Content, water content that cannot be extracted
through mechanical forces (e.g., from evapotranspiration or gravity);
θ(dθ/dψ= 0)
Water in the upper soil is generally in contact with air: θ < θsat i.e., unsaturated
soil
ψ : SOIL WATER POTENTIAL or SOIL WATER
SUCTION or SOIL WATER TENSION
[kPa] or [MPa] or [m] or [mm]
Water potential is the potential
energy of water per unit volume
relative to pure water in reference
conditions. Water potential
quantifies the tendency of water to
move from one area to another due
to osmosis, mechanical pressure, or
matric effects such as surface
tension.
Mechanisms for Water Retention
 directly related to surface tension (typically the most important), called
Matric or Tension or Capillary potential.
 related to particle charges, particles repelling each other, called Osmotic
potential
 chemical effect on clay particles that trap water particles (adsorption)
• We consider only the matric potential (capillarity)
    sat   0
Soil water potential at saturation is equal or very close to zero
ψ(θ): SOIL WATER RETENTION CURVE


 sat
r
Water Potential  [MPa]
-10
0
-10
-5

 sat
r

Silt
Sandy-Loam
Clay
Sand
FUNCTION OF THE SOIL TYPE
0
0.1
0.2
0.3
 [-]
0.4
0.5
Soil water content profile at equilibrium with water depth:
S e ,i Effective saturation
1
Se
DEGREE OF
SATURATION

S
 sat
Depth
EFFECTIVE SATURATION
 r
Se 
 sat   r
S e  [0  1]
Gravitational  ψ = 0 kPa -/- -33 kPa
Water in the
soil
  100
[kPa]
Capillary  ψ = -33 kPa -/- -3 Mpa
Hygroscopic (unavailable)  ψ < -3 Mpa; -10 Mpa
Free aspiration, possible because the water potential is
higher than absolute pressure.
  1500 /  4000
  33
[kPa]
[kPa]
Wilting point of plants. Usually plants are
not able to extract water below these
potentials. Wilting point is a plant dependent
property.
Field capacity is the the water potential that holds
water in the soil. Below field capacity gravity processes
(free drainage) become of minor importance. However,
it is a problematic definition.
At large water contents, water flows in the soil are controlled by gravity; at lower
water contents capillary is the dominating process with strong negative water
potentials.
SOIL HYDRAULIC HYSTERISIS


drying
drying
wetting
r
wetting

r
Wetting and drying boundary curves (main scanning curves)
 sat Saturation
 sat   Satiation
 sat   sat  


In experimental results and in common
discussion, no distinction is made
between Saturation and Satiation.
While hysteresis may be important and influence hydrological behavior, it is difficult
to model numerically and it is typically not accounted for.
FUNCTION FOR SOIL WATER RETENTION CURVES
   b S e 

1
0
Brooks and Corey, 1964
The parameters are dependent on the soil type
b
[kPa] Air entry soil water potential
1
1


  S e  m  1


Parameters:
1
n
0
[-] Pore-size distribution parameter
van Genuchten, 1980
1
m  1
n
n
[-]

[kPa-1]
INSTRUMENTS
The are several geophysical methods to measure soil water content in the field,
The most common soil moisture probes are using the Time-domain
reflectometry, TDR, principle.
The instrument to measure
water potential is called
“tensiometer” but typically
works at relatively high
water potentials
(> -200/-400 kPa)
WATER TRANSPORT IN THE SUBSURFACE
 The flow generally depends on water-vapor interactions, gravity and capillarity
forces, temperature gradients, but typically many of these complications are not
accounted for.
Darcy’s law is derived for saturated, homogenous, isothermal, isotropic media in a
steady-state
DARCY’s LAW [1856]
K sat  Ah3  h4   t
Vol
L
Hydraulic Head
h  z 
Gravity + Soil water potential
Darcy’s Law is generally expressed for unit area and in differential infinitesimal
terms in the vertical direction:
q   K sat
h
z
[m/s] or [mm/h]; Discharge per unit area
v
q

z
Real average velocity of the flow
Ksat = saturated hydraulic conductivity [m/s] or [mm/h]
More general when the system is tri-dimensional and anisotropic:
 

q   K  h

q  [q x , q y , q z ]
h: scalar

  
 [ , , ]
x y z
K x,x


K sat   K y , x
 K z , x
K x, y
K y, y
K z, y
K x,z 

K y ,z 
K z , z 
Principal axes:
h
q x   K xx
x
h
q y   K yy
y
h
q z   K zz
z
Darcy Law is an empirical generalization of the flow equation for laminar flow on
regular surfaces (e.g., Hagen–Poiseuille law); where viscous forces are
comparable with inertial forces, small Reynolds numbers (Re).

gd 2 h
q
32 x
q   K sat
h
z
Linear dependence of “flow” on gradients of hydraulic head.
Ksat = saturated hydraulic conductivity = function of fluid and pore size
distribution (porous media)
 Isotropic: KH = Kv
Anisotropic: KH ≠ Kv
Equal or different properties in different directions
aR 
KH
KV
Anisotropy ratio
 Homogenous:
K
0
z
Heterogeneous:
K
0
z
Spatial variability
of the properties
Averaging of hydraulic conductivity it depends on the geometrical composition of
the soil block and direction of flow (serial or parallel)
z1
z2
K1
K2
K1 z1  K 2 z2
KH 
z1  z 2
Simple
Mean
z1  z 2
KV 
 z1 z2 
 

 K1 K 2 
Harmonic
Mean
Hydraulic conductivity depends on domain size (it typically increases for larger
areas) this is due to:
•Macropores
•Preferential paths
•Spatial variability of properties
•Regions of low K can be by passed
Extension of Darcy’s Law to unsaturated flow (Buckingham, 1907)
K  K  
Unsaturated hydraulic conductivity –
Partially saturated hydraulic conductivity
As soil dries the connections between pores become irregular and discontinuous
θ ↓ K↓; Less connections, less water in the pores, more tortuous paths.
At higher water contents, K decreases more rapidly because larger pores are the
first to get empty. Almost all the water flow is transferred through big-pores.
This relationship is not hysteretic.
K
K sat

Unsaturated Conductivity K [mm h -1]
FUNCTION OF THE SOIL TYPE
10
0
10
-5
10
-10
Silt
Sandy-Loam
Clay
Sand
0
0.1
0.2
0.3
 [-]
0.4
0.5
SOIL HYDRAULIC CONDUCTIVITY CURVES
K  K sat S
 2

  3 

e  0 



K  K sat S e  1  1  S
0.5
Brooks and Corey, 1964

2
1/ m m
e
Parameters are soil type dependent
van Genuchten, 1980
VALUES OF HYDRAULIC CONDUCTIVITY
K sat
In Soil:
0.1 -500 [mm/h]
Gravel: 1 [m/s]
Rocks: 10-8 – 10-9 [m/s]
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103
[mm/h]
104 105
DARCY-BUCKINGHAM LAW (Unsaturated soil)



q   K    h
Principal axes:
h  z 

x

q y   K yy  
y

q z   K zz  
 K zz  
z
q x   K xx  
Diffusion Formulation of Darcy Law



q   K    h
D   K  
d
d





q   K      K    z
Soil Diffusivity





q   D     K    z
Conservation of Mass (Continuity Equation) + Conservation of Momentum
(Flow law = Darcy Law) = RICHARDS EQUATION
dV  dxdydz
q z
qz 
dz
z
qy 
qx
q y
y
dy
q x
qx 
dx
x
qy
qz
Mass
Conservation
in the dV
M  w

dV
t
t
RICHARDS EQUATION (1931)
Conservation of mass:
q x 


 w q x dydy   w  q x 
dx dydz  ...   w
dV
x 
t

q x q y q z




x
y
z
t
Flow equation:



q   K    h


q  
t





  K    h 
t
RICHARDS EQUATION (1931)




   K    h
t







   K      K    z
t

For isotropic, homogenous media in the vertical direction
  
 

  K  
 1 
t z 
 z

Equation that governs variably saturated subsurface flow
No general analytical solution is available
Solutions of Richards equation are demanding from a computational point of view (it
is a non-linear partial differential equation)
Numerical models are used to solve Richards equation (e.g., Hydrus-1D)
EXAMPLES
1  2
1  2
After some time
Horizontal
1
1
1
2
dz
2
 1   2  dz
1  2
After some time
2
1
1
2
dz
1  2
2
Vertical
PREFERENTIAL FLOW
Dual porosity – dual permeability model (Gerke and van Genuchten, 1993, 1996)
1-wf
Km
Vf/Vt = wf ≈ 0.01-0.05
Kf
Γw
Exchange term:
w  aw  f  m 
 f
  f
 w

 K f  
 1 
t
z
 z
 wf
 m 
w
  m 
 K m  
 1 
t
z
 z
 1  wf
Mass transfer coefficient:
[1 / h]
aw  aw K as , m , f 
K m , K f , sat , f , sat ,m , r ,m ,  f ,  m , n f , nm , w f , K as
Increasingly difficult to parameterize and to resolve numerically
[1 / mm h]
EXAMPLE: PREFERENTIAL FLOW
Ponded surface for 12 hours – clay-loam soil
0
-50
Time evolution: every 2
hours
Depth
Depth[mm]
(mm)
-100
-150
-200
No Preferential
flow
-250
Preferential flow
-300
0.25
0.3
0.35
0.4
[-]
θ [-]
0.45
0.5