Environmental Engineering Reference
In-Depth Information
a particular application (Hill and Tiedeman,
2007
). Nonlinear confidence intervals (Vecchia
and Cooley,
1987
) offer improved accuracy and
can be calculated with the above mentioned
programs, but only at considerable computa-
tional cost.
Monte Carlo methods use a known or
assumed distribution of one model parameter
value to infer information on the distribution of
model results or other model parameters. This
is accomplished through repeated application of
the model with values of the known parameter
randomly selected from the given distribution.
In groundwater-flow modeling, for example,
a log normal distribution, mean, and stand-
ard deviation of hydraulic conductivity may be
assumed for the simulated domain. Values of
hydraulic conductivity for the entire domain
(referred to as a realization) are randomly drawn
from that distribution. The groundwater-flow
model is solved for each realization to produce
estimates of recharge. If model runs are made
for 1000 realizations of hydraulic conductivity
fields, then the mean and standard deviation of
the 1000 calculated recharge fields can be deter-
mined at specific points in space and for the
entire simulated domain.
Monte Carlo methods can produce results
similar to those from inferential statistical
methods, and they have some advantage over
inferential methods when dealing with highly
nonlinear models (Hill and Tiedeman,
2007
).
Monte Carlo methods have been used with all
types of hydrologic models, including soil water-
budget models (Bekesi and McConchie,
1999
;
Baalousha,
2009
), watershed models (Bogena
et al
.,
2005
), and groundwater-flow models
(Hunt
et al
.,
2001
; Bakr and Butler,
2004
). Monte
Carlo methods are not limited to application to
models; they are also useful for assessing uncer-
tainty in recharge estimates derived by other
techniques, including water-budget methods
(Leake,
1984
; Dages
et al
.,
2009
).
aquifers. It is the dominant form of recharge
in many areas. Unsaturated zone water-budget
models can be useful tools for generating esti-
mates of diffuse recharge. There is great vari-
ability among these models, but a common
assumption is one-dimensional vertical water
flow through the unsaturated zone, so that a
study site can be represented by a single soil
column or by multiple noninteracting soil col-
umns. Two types of models are considered: soil
water-budget models and models based on the
Richards equation. The former type of model is
based on a water budget of the soil or root zone,
whereas the latter considers the water budget
over the entire thickness of the unsaturated
zone.
3.3.1 Soil water-budget models
Soil water-budget models for estimating
recharge usually consist of two components: a
water budget for the soil or root zone that esti-
mates the amount of drainage through the base
of that zone and a submodel that transports
that drainage to the water table. Drainage from
the soil zone,
D
, is typically determined from a
water-budget equation such as:
D
=− − −
P
ET
uz
∆
S
uz
R
(3.2)
off
where, following the notation of
Chapter 2
,
P
is
precipitation plus irrigation,
ET
uz
is evapotran-
spiration from the soil zone, Δ
S
uz
is change
in storage in the soil zone, and
R
off
is runoff.
Implicit in this approach is the assumption
that the maximum amount of storage within
the root zone is fixed, or that the depth of the
base of the root zone (more properly referred
to as the zero-flux plane,
Figure 2.2
) does not
vary with time. Equation (
3.2
) is usually solved
for daily values; the equation is valid for any
time-step size, including weekly, monthly, and
annual, but estimates of drainage may be sensi-
tive to time-step size. As discussed in
Section 2.2
,
annual drainage rates calculated with weekly
or monthly time steps are often less than those
calculated by using daily time steps.
Typical model input consists of daily pre-
cipitation and actual or potential evapo-
transpiration. Because measured values of
evapotranspiration are often not available,
some models allow input of air temperatures
3.3 Unsaturated zone water-
budget models
Diffuse recharge occurs as water moving
through the unsaturated zone to underlying
