Appendix 3 Green and Ampt model. Vertical infiltration

Appendix 3 Green and Ampt model. Vertical infiltration. This appendix shows the derivation of Equation 5.46 in detail. The logarithmic term of Equation 5.44, can be written as (5.45) where y = - -.Sf For y z < 1 , In (1 hf

+ y) can be expanded as

ln(1 + y ) = y - - +Y-2- - +Y+3 . . .y4 2 3 4

(A3.1)

Thus, Equation 5.44 becomes

et eisf 1 +-+-+... 2Sf sf 3hf 2hF

--2kthf

(

(A.3.2)

To find sf as a function o f t we first take the square root of Equation A3.2: t

-

(

(1

-;;;i)”2sf

In general, if P = a

2Sf sf +-+-+... 3hf 2hf

(A3.3)

+ bx + cx2 + dx3 + ...,then (A3.4)

Using this in Equation A3.3 and rearranging terms leads to: 112

7 + -36hf s? + .. . Again, in general, if Q = ax + bx2 + cx3 + . . ., where a # 0, then x = A Q + BQ2 + CQ3 + ...

( s t )

1

= S f + -sf 3hf

(A33

(A3.6)

1 b 1 where A = -, B = - -, C = - (2b2 - ac), etc. U a3 as This can be used to reverse Equation A3.5. The result will give sfas a function of t in series form:

where

fi

+

=[sr

Sf= f i t ” ? +f2t

+f3f3’? .. .

(5.46)

(5.47) (5.48)

(5.49)

This series is convergent for t smaller than a certain value, which depends on the value of the different coefficientsf,, f i , f 3 , etc.

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