# CE 394K.2 Hydrology

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```					     CE 394K.2 Hydrology

Infiltration
Reading for Today: AH Sec 4.3 and 4.4
Reading for Thurs: AH Sec 5.1 to 5.5

Subsequent slides prepared by Venkatesh
Darcy’s Law
• K = hydraulic conductivity
• q = specific discharge
• V = q/n = average velocity through the
area
h
Q   KA
L
Q     hdown  hup
q   K
A h       L
q z  K
z
Definitions
Element of soil, V
(Saturated)
V  gross volume of element                            Pore with
water
Vv  volume of pores                                        solid
Vs  volume of solids
Vw  volume of water
Vv                                                   Pore with
n   porosity                                          air
V
Vw
S     saturation; 0  S  1
Vv
V                                             Element of soil, V
  w  nS  moisture content; 0    n         (Unsaturated)
V
Infiltration
• General
– Process of water                                     
Saturation Zone
penetrating from                           
Transition Zone
ground into soil
– Factors affecting
• Condition of soil          Transmission
surface, vegetative        Zone
cover, soil properties,
hydraulic conductivity,
antecedent soil           Wetting Zone
moisture
– Four zones                                   Wetting Front
• Saturated,
depth
transmission, wetting,
and wetting front
Infiltration
• Infiltration rate        f (t )
– Rate at which water enters the soil at the surface
(in/hr or cm/hr)
• Cumulative infiltration
– Accumulated depth of water infiltrating during given
time period
t
F (t )   f ( )d
0

dF (t )
f (t ) 
dt
Infiltration Methods
• Horton and Phillips
– Infiltration models developed as approximate
solutions of an exact theory (Richard’s
Equation)
• Green – Ampt
– Infiltration model developed from an
approximate theory to an exact solution
Hortonian Infiltration
     
• Recall Richard’s                              D  K
t z  z 
Equation
    2 K
– Assume K and D are                        D 2 
t   z   z
constants, not a function
of  or z                                  2                       0
D 2
t   z
• Solve for moisture
diffusion at surface

f (t )  f c  ( f 0  f c )e kt

f0 initial infiltration rate, fc is constant rate and k is decay constant
Hortonian Infiltration
3.5

3        f0

2.5
Infiltration rate, f

k1
2
k1 < k2 < k3
1.5
k2

1
k3

0.5       fc

0
0                  0.5                   1     1.5   2
Time
Philips Equation
     
• Recall Richard’s                 D  K
t z  z 
Equation
– Assume K and D are
functions of , not z
• Solution
F (t )  St1/ 2  Kt
– Two terms represent
effects of                              1 1/ 2
• Suction head                 f (t )  St  K
2
• S – Sorptivity
– Function of soil suction
potential
– Found from experiment
Green – Ampt Infiltration
L  Depth to Wetting Front                                          h0
Ponded Water
 i  Initial Soil Moisture
Ground Surface

F (t )  L(   i )  L
Wetted Zone
L
dF      dL
f      
dt      dt
Wetting Front
h
q z  K    f
z                            i            

h  z
n

f K       K
z                       z                        Dry Soil
Green – Ampt                              Ground Surface


Infiltration (Cont.)                                  Wetted Zone            L

f K      K
z                                Wetting Front

• Apply finite difference to the                   i           

derivative, between                                       
– Ground surface z  0,  0                z                     Dry Soil
– Wetting front z  L,   f

 f 0                    f K      K
                                               z
f K    K K    K K        K
z       z        L0

F (t )  L
  f    
L
F                   f  K
 F      1

                                 
Green – Ampt                                      Ground Surface


Infiltration (Cont.)                                            Wetted Zone          L

  f                     dL

f  K        1
        f                                     Wetting Front
 F                         dt
i        
F (t )  L

dL     f   
       K
 L  1

dt                                               z                    Dry Soil

K            f dL                      Evaluate the constant of integration
dt dL 
           f L                       L  0 @t  0
Integrate                                  C   f ln( f )
K
t L  f ln( f  L)  C                                             f
                                        Kt L   f ln(                     )
 f L
Green – Ampt Infiltration                                   Ground Surface


(Cont.)                                                         Wetted Zone          L

f
Kt L   f ln(               )                               Wetting Front
 f L                              i        



F
F  Kt   f ln(1                 )               z
 f                                           Dry Soil

  f    
f  K
 F      1
             Nonlinear equation, requiring iterative solution.
          

See: http://www.ce.utexas.edu/prof/mckinney/ce311k/Lab/Lab8/Lab8.html
Soil Parameters
• Green-Ampt model requires
– Hydraulic conductivity, Porosity, Wetting Front
– Brooks and Corey
 r
se             Effective saturation
e
Soil Class   Porosity   Effective   Wetting    Hydraulic
e  n  r    Effective porosity                              Porosity     Front    Conductivity
Suction
  (1  se ) e                                      n          e                      K
(cm)        (cm/h)
Sand          0.437      0.417       4.95         11.78
Loam          0.463      0.434       9.89          0.34
Clay          0.475      0.385       31.63         0.03
Ponding time
• Elapsed time between the time rainfall
begins and the time water begins to pond
on the soil surface (tp)
Ponding Time

Infiltration rate, f
• Up to the time of ponding,                                             Potential
Infiltration

all rainfall has infiltrated (i
Rainfall
= rainfall rate)                                 i
f i           F  i *t p                                                            Actual Infiltration

  f                                                                                                      Time

f  K        1

Accumulated

Infiltration, F
Rainfall

Cumulative
 F                                                                                               Infiltration

  f     
i  K         1         Fp  i * t p
 i *t p    
           
 f
tp K
i (i  K )                                            tp                                        Time
Example
• Silty-Loam soil, 30%
 e  0.486
effective saturation,                                   16.7 cm
rainfall 5 cm/hr                                       K  0.65 cm / hr
intensity                                              se  0.30

  (1  se ) e  (1  0.3)(0.486)  0.340
  16.7 * 0.340
 f                          5.68
tp K                 0.65                            0.17 hr
i (i  K )            5.0(5.0  0.65)(i  K )

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