22
S. C. Lessoff and P. Indelman
The expressions (4), (6), and (7) detennine completely the flow parameters for a
single infiltration-redistribution cycle. It is possible to derive a solution for
intermittent regime of infiltrations and redistributions using the final conditions of
one infiltration redistribution cycle as the initial conditions for the next cycle.
However, the solution becomes complicated and this poses difficulties in further
statistical calculations. To simplify the solution for multiple cycles, we assume
that each cycle is long enough for the applied water to completely redistribute to
residual moisture content 6Ir • In our field test, applied water was always given at
least 4 days to redistribute before the next water application. Therefore, at the
beginning of each water application the field was nearly dry and flow had nearly
stopped. Under these conditions, the assumption of complete redistribution
produces results nearly indistinguishable from a solution that allows for
nonresidual moisture at the beginning of each irrigation cycle. By assuming
complete drying we can solve Eqs. (4), (6), and (7) for repeated cycles of
infiltrations and redistributions by replacing t~t-tk and W~Wk, where tk and Wk
are the starting time and the applied water of cycle k (Lessoff et al. 2001).
3.1.2 Transport
For the purpose of simulating field-averaged transport, we assume that advection,
equilibrium sorption, and exponential decay govern transport in a single tube, and
we neglect the effect of pore scale dispersion. The solute concentration C(z,t)
(expressed as total mass of solute/total volume of soil) obeys the conservation
equation:
(8)
In Eq. (8), u(z,t) is the local solute velocity defined by:
u(z, t)=q(z, t)/ R( (1)
(9)
where q is defined by Eqs. (4), (6), and (7), Kd is the partition coefficient, and Pb
is the soil bulk density. The solution to Eq. (8) for a narrow pulse initially starting
from the soil surface z=o was derived by Indelman et al. (1998). To detennine the
solute concentration for multiple infiltration-redistribution cycles we need the
solution of Eq. (8) for a solute pulse initially located at any depth. The exact
solution ofEq. (8) subject to the initial condition C(z, 0) = MoO{z-zo) is
C(z,t)=Moe- A1 xCt)8{[z-zr(t)] xCt)},
(10)
with xCt)=[6Ilt;)+KdPb]/[6Ilt)+KdPb]' The function X(t) should be regarded as a
stretching of a pulse of unit area along the z-axis. Front moisture content 6Ilt) is
defined by Eqs. (6) and (7). The location of the solute pulse zr(t) was derived by
Lessoff et al. (2001) as follows:
S. C. Lessoff and P. Indelman
The expressions (4), (6), and (7) detennine completely the flow parameters for a
single infiltration-redistribution cycle. It is possible to derive a solution for
intermittent regime of infiltrations and redistributions using the final conditions of
one infiltration redistribution cycle as the initial conditions for the next cycle.
However, the solution becomes complicated and this poses difficulties in further
statistical calculations. To simplify the solution for multiple cycles, we assume
that each cycle is long enough for the applied water to completely redistribute to
residual moisture content 6Ir • In our field test, applied water was always given at
least 4 days to redistribute before the next water application. Therefore, at the
beginning of each water application the field was nearly dry and flow had nearly
stopped. Under these conditions, the assumption of complete redistribution
produces results nearly indistinguishable from a solution that allows for
nonresidual moisture at the beginning of each irrigation cycle. By assuming
complete drying we can solve Eqs. (4), (6), and (7) for repeated cycles of
infiltrations and redistributions by replacing t~t-tk and W~Wk, where tk and Wk
are the starting time and the applied water of cycle k (Lessoff et al. 2001).
3.1.2 Transport
For the purpose of simulating field-averaged transport, we assume that advection,
equilibrium sorption, and exponential decay govern transport in a single tube, and
we neglect the effect of pore scale dispersion. The solute concentration C(z,t)
(expressed as total mass of solute/total volume of soil) obeys the conservation
equation:
(8)
In Eq. (8), u(z,t) is the local solute velocity defined by:
u(z, t)=q(z, t)/ R( (1)
(9)
where q is defined by Eqs. (4), (6), and (7), Kd is the partition coefficient, and Pb
is the soil bulk density. The solution to Eq. (8) for a narrow pulse initially starting
from the soil surface z=o was derived by Indelman et al. (1998). To detennine the
solute concentration for multiple infiltration-redistribution cycles we need the
solution of Eq. (8) for a solute pulse initially located at any depth. The exact
solution ofEq. (8) subject to the initial condition C(z, 0) = MoO{z-zo) is
C(z,t)=Moe- A1 xCt)8{[z-zr(t)] xCt)},
(10)
with xCt)=[6Ilt;)+KdPb]/[6Ilt)+KdPb]' The function X(t) should be regarded as a
stretching of a pulse of unit area along the z-axis. Front moisture content 6Ilt) is
defined by Eqs. (6) and (7). The location of the solute pulse zr(t) was derived by
Lessoff et al. (2001) as follows: