M
Z
Quasi Steady-State SimulaƟon of the
Unsaturated Zone in Groundwater Modeling
of Lowland Regions
S
S
:V
Paul E. V. van Walsum* and Piet Groenendijk
Well-conceived and detailed simula on of soil-moisture processes is a prerequisite for accurate watershed-scale
modeling of water quan ty and quality processes. For this purpose, Richards’ equa on (and its extensions) is the
conceptually preferable op on. Applying the equa on on the watershed scale, however, may overstretch available
computer resources. At the other extreme, methods based on lumping are oversimplified. Approaches are therefore
needed that are efficient and just accurate enough, and that provide the required detail in the ver cal column. We
have developed a quasi-steady-state model that uses a sequence of steady-state water content profiles for performing dynamic simula ons. The appropriate profiles are—for each me level—selected on the basis of water balances at
the aggregate scale of control volumes. The groundwater coupling scheme involves an itera on cycle for the phrea c
storage coefficient. In the postprocessing stage, the values of state variables obtained using the coupled model are
disaggregated, thus delivering pressure heads, moisture contents, and fluxes at the detailed scale of compartments
of a Richards-type model. The plausibility of the simplified approach was tested by comparing its results to those of
a Richards-type model. The results appear promising for at least three-quarters of the area of the Netherlands with
a shallow groundwater eleva on (within 2 m of the soil surface) and a thin root zone (<0.5 m thick). Customizing the
modeling method used to the situa on conserves computa onal resources, allowing more room for doing sensi vity
analyses. This could be instrumental for quan fica on of model reliability.
A
: ME, model efficiency coefficient.
A
of groundwater elevation dynamics is necessary for simulating unsaturated–saturated zone
interactions and drainage flow, for solving inverse modeling
problems that use results of groundwater monitoring and remote
sensing, for simulating water quality processes, and for performing
agroecological evaluations of the hydrologic regime. The shallower
the groundwater elevations, the more important it is to simulate
them accurately. In a lowland region like the Netherlands, the
groundwater elevation is within 2 m of the soil surface in 85% of
the area. Subtle variations of soil surface elevation—on the order
of 0.2 m—then can have a significant impact on local hydrologic
conditions. In the past decade, soil elevation data have become
available at the resolution of 5 by 5 m2, and land use data at the
scale of 25 by 25 m2. This has helped in creating demand for
high-resolution hydrologic modeling, even though the availability
of data on soil hydraulic properties has lagged behind. In parallel, there is increasing demand for uncertainty analyses and for
lengthy simulations to estimate climate-related statistics like the
frequency distribution of saturation-generated surface runoff.
Since the publication of Freeze (1971), a number of models
have become available for the simulation of (quasi) three-dimensional, transient water flow in variably saturated porous media.
Examples are SUTRA3D (Voss and Provost, 2002), FEMWATER
(Lin et al., 1997), HYDRUS-3D (Šimůnek et al., 2006), LGMSWAP (Stoppelenburg et al., 2005), MODFLOW-VSF (Thoms
et al., 2006), HydroGeoSphere (Sudicky et al., 2006), and PIHM
(Duffy, 1996, 2004; Qu, 2004). Although these models vary in
specific details, all are based on the use of Richards’ equation
to describe both saturated and unsaturated flow. The drawback
of these approaches is that they impound heavily on available
computational resources. For instance, an example application of
MODFLOW-VSF, involving 2500 soil columns, runs 45 min for
a year of simulation on a standard PC. That is one or two orders
of magnitude too slow for our purposes: a model is needed that
can be run with half a million units for a time series of 30 yr on
a single standard PC.
For large-scale simulations, modeling systems like MIKE
SHE (Refsgaard and Storm, 1995), MODFLOW-UZF1
(Niswonger et al., 2006), HBV (Bergström, 1976, 1995), and
SWAT (Arnold et al., 1998; Neitsch et al., 2002) offer simplified one- or two-layer modeling of the unsaturated zone. But for
our purposes, these approaches nonetheless fall short in three
ways. First, most of them are based on “lumping,” meaning
that moisture-content variations within the layers themselves
remain a black box. Second, many approaches use an oversimplified method for simulating the capillary rise from the phreatic
Alterra, Wageningen Univ. and Research Centre, P.O. Box 47, 6700AA
Wageningen, the Netherlands. Received 22 Aug. 2007. *Corresponding
author ([email protected]).
Vadose Zone J. 7:769–781
doi:10.2136/vzj2007.0146
© Soil Science Society of America
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All rights reserved. No part of this periodical may be reproduced or
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www.vadosezonejournal.org · Vol. 7, No. 2, May 2008
769
Steady States
surface to the root zone. For instance, MODFLOW-UZF1 uses
the extinction depth concept, which takes into account only the
depth to the phreatic surface. The water content of the root zone
is just used for checking whether there is a water deficit but not
for computing the capillary rise itself. Third, all of the approaches
use a constant phreatic storage coefficient. The simulation of the
dynamics of shallow water tables is then unreliable, as is the simulated drainage flow to the surface water.
In our opinion, the conceptual gap between the existing simplified approaches and those using Richards’ equation is large
and needs to be filled. It appears that this can be done by making
creative use of presolved water content profiles that comply with
the steady-state form of Richards’ equation. We have developed
a quasi-steady-state model that uses a sequence of steady-state
profiles for performing dynamic simulations. The appropriate
profiles are—for each time level—selected on the basis of water
balances at the aggregate scale of control volumes. The groundwater coupling scheme involves an iteration cycle for the phreatic
storage coefficient. The method described is a radical redesign of
the one presented by De Laat (1976, 1980, 1992), which was a
further development and computerized implementation of ideas
presented by Wesseling (1957) and Rijtema (1965).
Computational efficiency is an essential aspect of the method.
Therefore, the description of the method is followed by a summary of its numerical implementation; a step-by-step numerical
example goes along with it. Because there is an internal contradiction in using steady-state profiles for dynamic modeling, the
method contains steps that do not directly follow from mathematical derivations. Partly for this reason, we present results of
a “plausibility test”; criteria are used for comparing the results
with those obtained from a Richards-type model for 21 soil types.
We point out the main differences with the method of De Laat
(1976). Then some further results are presented, showing how the
model reacts differently than a Richards-type model to dynamic
top-boundary conditions. Finally, we make a statement about the
operationality of the method, followed by an outlook with respect
to its possible role in large-scale groundwater modeling.
For one-dimensional flow in an unsaturated soil with root
water extraction, the steady-state form of the flow equation can
be written as
⎛ dψ
⎞⎤
d ⎡
⎢ K ( ψ)⎜⎜
+ 1⎟⎟⎟⎥ −τ ( ψ, z ) = 0, 0 ≥ z ≥ h
⎝ dz
⎠⎥⎦
d z ⎢⎣
subject to the boundary conditions
ψ (h ) = 0
[1b]
⎡
⎛ dψ
⎞⎤
⎢ K ( ψ )⎜⎜
+ 1⎟⎟⎟⎥
= −q ( 0 )
⎢⎣
⎝ dz
⎠⎥⎦ z =0
[1c]
where z is the elevation coordinate [L], taken positively upward
(and zero at the soil surface), h is the groundwater elevation [L],
ψ is the pressure head [L], K(ψ) is the hydraulic conductivity [L
T−1] as a function of pressure head (Mualem, 1976), q(0) is the
flux density at the soil surface [L T−1], taken positively upward,
and τ(ψ,z) represents a depth- and head-dependent extraction
term for root water uptake [L3 L−3 T−1], like that given by
Kroes and van Dam (2003). A steady-state profile is obtained
by specifying the conductivity parameters of each distinguished
soil layer and by solving Eq. [1] subject to imposed values for
the groundwater elevation h, the potential flux density at the soil
surface qpot(0), and the potential total root water uptake rate Tpot
of the root zone. A flexible root distribution function with depth
is applied, yielding the potential root extraction rate τpot(z). The
actual root extraction rate is obtained (as part of the solution
scheme for Eq. [1]) through multiplying the potential extraction
rate by a dimensionless reduction function (Feddes et al., 1978).
The solution obtained by running a steady-state version of the
SWAP model (Kroes and van Dam, 2003) yields values for the
actual root extraction rate τact(z), the actual flux density q(0) at
the soil surface, and the constant moisture flux density q in the
subsoil below the root zone. The flow in the saturated part of
the profile is outside the domain of the steady-state simulations.
Examples of steady-state profiles for a root zone thickness of 0.3
m and a groundwater elevation h = −1.5 m are given in Fig. 1.
The set of steady-state profiles that result from different
combinations of h, qpot(0), and Tpot is assumed to be available
in a database. For the sake of simplicity, the database only contains steady states resulting from either a non-zero qpot(0) < 0
or a non-zero Tpot > 0 (plus the equilibrium profiles). From the
states contained in the database, it is then possible to construct
a function Ψ(z,q,h) for the pressure head [L] as a function of
the elevation z [L], the steady-state flux density q below the root
zone [L T−1], and the groundwater elevation h [L]. Combining
Ψ(z,q,h) with a relationship between moisture content and pressure head (van Genuchten, 1980) yields the function Θ(z,q,h)
for the volumetric water content [L3 L−3] as a function of z,
q, and h.
In utilizing the profiles, frequent reference is made to a profile segment. By a root zone profile segment is meant the part of a
profile for z ≥ zr, where zr is the elevation of the bottom of the
root zone. By a subsoil segment is meant the part of a profile for
z < zr, extending downward to an elevation zs (which is taken
below the deepest groundwater elevation that can locally occur).
Modeling Method
The model schematization assumes that unsaturated flow
takes place within parallel, vertical columns, with each column
connecting to a simulation unit of a groundwater model. The
phreatic surface acts as a “moving boundary” between the flow
domains of the soil column models and the groundwater model.
All lateral exchanges are assumed to take place in the saturated
zone, which are described by the groundwater model. The main
idea of the modeling method for the (unsteady) unsaturated
flow is to use steady-state solutions to Richards’ equation as
building blocks of a dynamic model, a so-called quasi-steadystate model. The appropriate building blocks are—for each time
level—selected on the basis of water balances at the aggregate
scale of control volumes for the “root zone” and the “subsoil.”
Put in mathematical terms, the partial differential equation for
the unsteady flow (Richards’ equation) is replaced by two ordinary differential equations: one for the variations in the vertical
column (using the steady-state form of the flow equation) and
the other for the variations in time (using a water balance at the
aggregate scale).
www.vadosezonejournal.org · Vol. 7, No. 2, May 2008
[1a]
770
Transi ons between Steady States
In solving the equations describing the soil water dynamics,
use is made of the following properties of the Θ(z,q,h) function:
1. For a given groundwater elevation, the total amount of water
s in a steady-state profile is a strictly monotonously decreasing
function of the steady-state flux q (taken positively upward).
The same applies to the total amount of water present in the
profile from the groundwater elevation upward.
2. For a given moisture content at the top of the profile, the
total amount of water in the profile is a strictly monotonously
increasing function of the groundwater elevation h.
To describe the method, first two hypothetical cases of a
drying soil are considered. Both start from a steady-state moisture profile involving root zone extraction (meaning a negative
recharge rate R) and capillary rise. The cases differ with respect to
the boundary conditions starting from time t j, where the superscript j denotes the time level.
In Case 1 (Fig. 2), the extraction suddenly increases to a new
value T j+1 that is assumed to be known; the groundwater elevation is held constant by supplying sufficient lateral saturated flow
to compensate for the capillary rise. The schematization of the
quasi-steady-state method entails that the water content profile
at t j+1 has to be found by selecting one of the profiles from the
database of steady states. The unknown to be solved is the (new)
steady-state flux density at t j+1. If that flux density is known, then
(in combination with the known groundwater elevation h) the
rest of the profile is known. The solution is found by formulating
a water balance equation for the vadose zone as a system volume,
assuming that the value of q in the Θ(z,q,h) function occurs at
the phreatic surface:
F . 1. Examples of steady-state profiles for a loamy soil with a root
zone thickness of 0.3 m and a groundwater eleva on of −1.5 m. For
the capillary rise profiles (transpira on rate T > 0) the given values
of the flux density (q > 0) are for below the root zone; for the equilibrium profile and the percola on profiles (infiltra on rate I ≥ 0)
the given values of the flux density (q ≤ 0) are for the whole profile
down to the groundwater eleva on. The Mualem–van Genuchten parameters (Mualem, 1976; van Genuchten, 1980) given by
Wösten et al. (2001) for this homogeneous soil are saturated volumetric water content θs = 0.41 m3 m−3, residual volumetric water
content θr = 0.01 m3 m−3, saturated hydraulic conduc vity Ks =
0.0370 m d−1, and the empirical shape parameters α = 0.71 m−1, n
= 1.298, and λ = 0.912.
Water content totals of the profile segments are obtained through
integration:
0
sr = ∫ θ ( z ) d z ; ss = ∫
zr
zr
zs
θ ( z )dz ;
0
s = ∫ θ ( z )dz
[2]
0
∫h
zs
where θ(z) is the volumetric water content [L3 L−3] at elevation
z [L], sr is the total water content of the root zone [L], ss is the
total water content of the “subsoil” [L], and s is the total of sr
and ss [L].
⎡ θ j +1 ( z ) − θ j ( z )⎤ d z − q haveΔ t = R aveΔ t
⎢⎣
⎥⎦
θ j +1 ( z ) = Θ( z , q hj +1 , h )
[4a]
[4b]
Dynamics
Recharge
Flow dynamics are driven by time variations of water balance stresses in the root zone. These stresses are summarized by
the recharge rate:
0
R = I − E − ∫ τ act ( z ) d z
zr
[3]
where R is the recharge rate [L T−1], I is the infiltration rate at
the soil surface [L T−1], E is the evaporation rate [L T−1], and
τact(z) is the actual root extraction rate as a function of z [L3 L−3
T−1]. For the evaporation, a simplified approach according to
Boesten and Stroosnijder (1986) is used. The head-dependent
function for root extraction is nonlinear (Feddes et al., 1978); it
is therefore important for accuracy to evaluate the dependency at
a detailed scale in the vertical column. The use of a total recharge
rate implies that infiltration at the soil surface is assumed to be
distributed instantaneously throughout the root zone according
to one of the steady-state water content profiles. It is part of the
solution procedure to determine which profile that is.
www.vadosezonejournal.org · Vol. 7, No. 2, May 2008
F . 2. Case 1 of a drying soil: Increasing extrac on from the root
zone (transpira on rate T j+1 > T j) and the groundwater eleva on
h held constant. The simulated transi on from the steady-state
profile at me t j to a steady-state profile at t j+1 is based on a water
balance at aggregate scale, using the me-averaged transpira on
rate Tave, the total volume change ∆s, and the me-averaged flux
density qhave from the phrea c surface.
771
where Rave is the time-averaged recharge rate (which in Case 1
is equal to −T j+1) [L T−1], qhj+1 is the flux density through the
phreatic surface at time t j+1 [L T−1], qhave is the time-averaged
flux density during the interval (t j, t j+1) [L T−1], and Δt is the
time increment (t j+1 − t j) [T]. The time-averaged flux density is
computed with a weighting factor f (dimensionless) for the time
dependence:
q have = (1− f )q hj + fq hj +1
θ j +1 ( z ) = Θ( z , q rj +1 , h ),
0
0
∫h θ
j +1
[5]
j
( z ) d z − fq hj +1Δ t =
( z ) d z + (1− f
)q hj Δ t
zr
∫h
[6]
+ R aveΔ t
0
j +1
( z ) − θ j ( z )⎤⎥ d z − q raveΔ t = R ave Δ t
⎦
z < zr
[8a]
[8b]
The used general principle of locating the q in the Θ(z,q,h) function (here at the phreatic surface) is to place it at the bottom of
the profile segment considered, as far away as possible from the
dynamic top boundary condition.
The appropriate groundwater elevation depth below the root
zone (denoted by dc) at which the switch is made to the alternative method (using a composite profile) is soil dependent and
must be found through calibration on a Richards-type model.
Such a composite profile is by its very nature dynamic, due to
the imbalance between the fluxes in the separate segments. If the
hypothetical example was continued from t j+1 by reducing the
root extraction in such a manner that the root zone water content remains stationary from then on, then the subsoil segment
would slowly converge to the root zone one, eventually forming
one continuous profile.
In Case 2 (Fig. 4), the moisture content at the top of the
profile is kept constant. That is done by slowly reducing the
extraction. This compensates for the decrease in capillary rise due
to the falling groundwater elevation. The lateral saturated flow is
assumed to be zero. In this (hypothetical) case, the extraction rate
as a function of time is assumed to be known. The unknown to
be solved is the groundwater elevation as a function of time. That
[7a]
F . 3. Use of a so-called composite profile in the alterna ve
solu on method for situa ons with capillary rise and deep groundwater eleva ons, applied to the hypothe cal Case 1 of a drying
soil (cf. Fig. 2). In the first step of the solu on procedure, a water
balance for the root zone is used (with ∆sr as the total volume
change) to obtain the profile segment at me t j+1, yielding also the
me-averaged flux through the bo om of the root zone, qrave. In
the second step, this flux density is then used as a top boundary
condi on in the water balance of the subsoil (with ∆ss as the total
volume change), yielding the steady-state profile segment for the
subsoil and the flux density through the phrea c surface at t j+1.
www.vadosezonejournal.org · Vol. 7, No. 2, May 2008
⎡ θ j +1 ( z ) − θ j ( z )⎤ d z − q haveΔ t = −q raveΔ t
⎢⎣
⎥⎦
θ j +1 ( z ) = Θ( z , q hj +1 , h ),
From Property 1 of the set of steady states, it follows that the
left side of Eq. [6] (in combination with Eq. [4b]) is a strictly
monotonously decreasing function of the steady-state flux at t j+1.
For a given value of the right side of Eq. [6], it is possible to scan
the set of steady states and then find the (only) one that satisfies the equation. This yields the unknown qhj+1 and thereby (in
combination with h) the whole water content profile by inserting
the values in the Θ(z,q,h) function.
For deep groundwater levels and a drying soil, it is more
realistic to let the water content profile in the subsoil lag behind
that of the root zone. In that case, separate segments of water
content profiles are used for the root zone and the subsoil (Fig.
3); together the segments form a so-called composite profile. The
profile segment of the root zone is found through a reformulation of Eq. [4], now assuming that the value of q in the Θ(z,q,h)
function occurs at the root zone boundary (so the integration of
θ starts from zr):
∫zr ⎢⎣⎡ θ
[7b]
where qrj+1 is the flux density through the root zone boundary
at time t j+1 and qrave is the time-averaged flux density [L T−1].
The method for solving the equations is similar to the one used
for Eq. [4]. The obtained qrave can subsequently be used as a
boundary condition in the water balance of the subsoil. In this
water balance, the value of q in the Θ(z,q,h) function is assumed
to occur at the phreatic surface, as in Eq. [4a] (but now with the
integration of θ ending at zr):
Then, after rearranging, Eq. [4a] can be written as
∫h θ
z ≥ zr
F . 4. Case 2 of a drying soil: Moisture content at the top of the
profile held constant and a falling groundwater eleva on due to
capillary rise. A water balance involving the me-averaged transpira on rate Tave, the total volume change ∆s, and the saturated
flow to the soil column G is used for simula ng the transi on to
a groundwater eleva on h j+1, assuming that the water content
profile at me t j+1 is a steady-state profile.
772
is done by formulating a water balance condition for a control
volume of the soil column extending into the groundwater:
0
∫zs ⎡⎢⎣ θ
j +1
( z ) − θ j ( z )⎤⎥ d z = (R ave + G ave )Δ t
⎦
Step 2 deviates from the method for Case 2 only in the handling
of the moisture conditions in the top part of the profile, as is
explained below. For simulating situations with percolation, the
alternative method is always used (Eq. [7] and [8]), involving
composite profiles as shown in Fig. 3. The flow chart in Fig. 6
summarizes the overall solution procedure.
[9a]
θ j +1 ( z ) = Θ (z , q j +1 , h j +1 )
[9b]
θ j +1 ( 0 ) = θ j ( 0 )
[9c]
Flux Density Profile
The flux density profile can be made explicit through making
a water balance for each desired elevation coordinate zi:
q ave ( z i ) =
[10]
z
G ave − ∫ i ⎢⎣⎡ θ j +1 ( z ) − θ j ( z )⎥⎦⎤ Δ t + τ ave
act ( z ) d z
zs
where the saturated flow G to the soil column [L T−1] is assumed
to be zero in this hypothetical case. This equation uniquely determines the groundwater elevation at t j+1, owing to Property 2 of
the set of steady states: Eq. [9a] can be rearranged as was done
with Eq. [4a], resulting in an equation with the unknown θ j+1on
the left side. The solution at t j+1 is then found by scanning the
subset of steady states that complies with Eq. [9b] and [9c] and finding the (only) one that satisfies the rearranged form of Eq. [9a].
The general case (Fig. 5) involves both a groundwater elevation change and a change in the water content at the top of the
profile. To find a unique solution in a straightforward manner,
the transition of the profile from t j to t j+1 is partitioned. This is
done with the moisture profile for the groundwater elevation at
t j and the moisture content at the top of the profile at t j+1, as
indicated in Fig. 5 by the curve at t~j+1. The partitioning of the
water-content change is based on the notion that the recharge of
the root zone acts as a driver of the processes in the column; thus
the groundwater elevation change is a reaction to what happens
above. Furthermore, this way of partitioning greatly simplifies
the solution method, which then can consist of two major steps
that are performed consecutively:
1. For the unchanged groundwater elevation, determine the
intermediate update at t~j+1.
2. For the new moisture conditions in the top part of the profile,
determine the new groundwater elevation at t j+1.
The solution method used for Step 1 is the same as for the
hypothetical Case 1 described above. In this step, the elevation of
the phreatic surface is assumed unchanged; thus the elevation is
used in an explicit manner with respect to time. The method for
{
}
where qave(zi) is the time-averaged flux density [L T−1], Gave is
the time-averaged saturated flow to the column [L T−1], and
τactave(z) is the time-averaged actual root extraction rate [L3 L−3
T−1]. The term involving θ is the so-called volume change rate.
This causes the time-averaged “quasi-steady-state” flux densities
to differ from the steady-state flux density of a steady-state water
content profile. The computed fluxes for elevations below the
phreatic surface are outside the flow domain of the soil column
model. For accurate simulation of water quality processes, they
should be replaced by fluxes that are determined in the groundwater model.
Model ImplementaƟon
For the method to be efficient, it is crucial that the computational effort for running a regional hydrologic model
(describing three-dimensional groundwater flow coupled to the
one-dimensional flow in the unsaturated zone) should be kept
F . 5. General case of a drying soil, involving both a water conF . 6. Flow chart of solu on methods used. The groundwater
tent change at the top of the profile and a groundwater eleva on
change: The moisture profile at me t = t~j+1 is used for “par oning” the water content change from t j to t j+1. The profile at t~j+1 is
denoted as the intermediate update.
www.vadosezonejournal.org · Vol. 7, No. 2, May 2008
eleva on depth dc below the bo om of the root zone (zr) is
assumed to be available from calibra on on a Richards-type model
in the preprocessing.
773
to a minimum. To that end, the numerics are done, insofar as
possible, in the pre- and postprocessing stages.
Since an explicit scheme is used for handling some of the variables with respect to time, the accuracy of the method is sensitive
to the time increment used. Step 3 contains special precautions to
ensure the stability of the scheme. A numerical example is used
for explaining the computational steps. For the example, we use
the soil with a root zone thickness of 0.30 m that was previously
drawn on in the illustrations. The initial groundwater elevation
h at time t j is −1.5 m and the moisture profile in the vadose zone
is at equilibrium. A time step of 1 d is used.
Preprocessing
In the preprocessing stage, steady states for each soil type and
possible root zone depth are computed for the following:
• a series of (potential) boundary flux values for the root zone,
ranging from extreme potential infiltration to extreme potential evapotranspiration
• a series of groundwater elevations, ranging from just below the
soil surface to the deepest depth present in the study region
The used soil type definitions may involve several soil layers
having different hydraulic properties. For each of the computed
steady-state profiles, the mean pressure head in the root zone is
determined with
1 0
ψ( z )dz
[11]
z r ∫zr
where zr is the elevation of the bottom of the root zone (m)
and ψ(z) is the pressure head as a function of z in one of the
steady-state profiles (m). Each steady-state profile has a unique
combination of ψr and groundwater elevation h. These two variables serve as entries in tabular functions. This deviates from the
use of the flux density q as an “independent” variable in the
functions Θ(z,q,h) and Ψ(z,q,h) that are used in describing the
method. In the implementation of the model, the flux density is
handled as a tabular function, tbq(ψr,h). Using ψr instead of q as
an independent variable appears to be more convenient for the
algorithmic part of the implementation. The function Ψ(z,q,h) for
the pressure head is implemented as the set of tabular functions
tbΨi(ψr,h) for all compartments i of the Richards-type model
used in the preprocessing. The function Θ(z,q,h) for the water
content is implemented as tbΘi(ψr,h), and also at the aggregate
scale of the used control volumes, using the integrations given
in Eq. [2]:
• tbsr(ψr,h) for the total storage of water in the root zone
• tbss(ψr,h) for the total storage of water in the subsoil
ψr =
F . 7. Example of a tabular root zone storage func on sr(ψr,h) for
a loamy soil with a root zone thickness of 0.30 m (for Soil Type 21
of Table 2; the parameters are given in the cap on of Fig. 1). The
total storage in the root zone is a func on of the mean pressure
head in the root zone (pF of ψr) and the groundwater eleva on (h).
Examples of a storage and a flux-density function are given
in Fig. 7 and 8. (Note that the pF axes of the two figures are different, for graphical reasons.) The steady-state simulations only
yield values for physically possible combinations of ψr and h,
so there is only partial independence of these two variables. To
make possible a step-by-step solution procedure, the functions
are extended to the whole domain of ψr and h, simply based on
the rule “groundwater prevails.”
Online Computa onal Scheme
The online part of the calculations is done in combination
with a regional hydrologic model, with feedback at each time step;
the scheme involves three major steps:
1. Calculate the recharge of the root zone.
2. Update the root zone pressure heads of the moisture profile
segments (Step 2a for the root zone update and Step 2b for
the subsoil update), yielding the values for the intermediate
solution step at t~j+1.
3. Update the groundwater elevation in conjunction with a
groundwater model; finalize root zone pressure heads, yielding the values at t j+1.
www.vadosezonejournal.org · Vol. 7, No. 2, May 2008
F . 8. Example of a tabular moisture flux density func on q(ψr,h)
for a loamy soil with a root zone thickness of 0.30 m (for Soil Type
21 of Table 2). The steady-state flux density is given as a func on of
the mean pressure head in the root zone (pF of ψr) and the groundwater eleva on (h).
774
Recharge
1. Example of a tabular combined storage and me-integrated flux func on [sr − qΔt](ψ r) that is used in the root zone
update of the pressure head ψ r for elapsed me Δt = 1 d. The
func on values of q(ψ r,h) (moisture flux density) and s(ψ r,h)
(storage in the root zone) are obtained from the func ons
depicted in Fig. 7 and 8 for a groundwater eleva on of h = −1.5 m.
T
The recharge of the root zone (Step 1) is computed as the
net balance of the following three components: infiltration at
the soil surface, evaporation, and root extraction (Eq. [3]). The
mean pressure head of the root zone is disaggregated to a more
detailed level of compartments i (which are also used in solving
the steady-state form of Richards’ equation) by evaluating the
functions tbΨi(ψr,h); then more accurate results are obtained
for the total root water uptake. The time-averaged recharge rate
is set equal to the value that is obtained using state variables (ψi)
at the beginning of the time step, so an explicit scheme is used.
In the numerical example, there is a recharge rate of R ave = I ave
= 0.016 m d−1 for the time step from t j to t j+1.
pFr of ψ r
q(ψ r,h)
m d −1
0.0000
sr(ψ r,h)
[sr – qΔt](ψ r)
————————m ————————
2.13
2.00
−0.0007
0.1058
0.1100
0.1058
0.1107
1.80
−0.0020
0.1149
0.1169
1.60
−0.0040
0.1182
0.1222
1.40
−0.0061
0.1201
0.1262
…
Root Zone Pressure Head
In Step 2a of the solution scheme, ψr (the mean pressure
head of the root zone) is updated. This yields the value for the
intermediate solution step, ψr~j+1. In the initial situation, the
equilibrium pressure head varies from −1.5 m at the soil surface
to −1.2 m at the bottom of the root zone; so the mean value is
equal to −1.35 m, which corresponds to a pF value of 2.13. For
this value in combination with h = −1.5 m, the tabular function
displayed in Fig. 7 gives a total storage in the root zone of srj =
0.1058 m. According to the model concept, the infiltration water
(0.016 m) will be spread throughout the root zone, so it is certain that the model will simulate percolation from the root zone
during this time interval. In that case the “alternative” solution
method is used, involving separate profile segments for the root
zone and the subsoil, as indicated in the flow chart of Fig. 6. Then
Eq. [7a] gives the water balance condition for the transition to
the intermediate update at t~j+1. A fully implicit scheme for the
time weighting of the moisture flux is used in this example, so
qrave = qr~j+1. By inserting this into Eq. [7a], using the expression
for sr given in Eq. [2], and rearranging, the balance is obtained
in the following form:
0 ~ j +1
∫zr θ
After the update of the root zone variables, the flux to the
subsoil is available for making the balance given in Eq. [8a].
Thereafter, a similar process of matching water availability and
water demand serves as the solution procedure for the pressure
head of the subsoil segment. Only the result is mentioned here.
Assuming the bottom of the subsoil control volume is at zs = −2.0
m, then the initial storage ssj in the subsoil equals 0.6668 m. The
result for the intermediate update (Fig. 9) is a storage of ss~j+1
= 0.6702, which implies a change of Δss~j+1 = 0.0034 m and a
time-averaged flux to the phreatic surface of −qhave = 0.0004 m
d−1. The new total storage in the column is given by s~j+1 = sr~j+1
+ ss~j+1 = 0.1180 + 0.6702 = 0.7882 m.
Coupling to a Groundwater Model
The coupling to a groundwater model involves passing of
information concerning the groundwater recharge and the storage characteristics of the vadose zone for each simulation unit.
The model receives in return the groundwater elevations and the
totals of the saturated flow to the vertical columns. The coupling
involves the implementation of Eq. [9], but from a different
starting point: the hypothetical case starts from t j; here the start
0
( z ) d z − q r~ j +1Δ t = ∫ θ j ( z ) d z + R ave Δ t
zr
[12]
= s rj + R aveΔ t = 0.1058 + 0.0160 = 0.1218 m
The terms on the left side are now replaced by the respective
tabular functions, with the unknown root zone pressure head
ψr~j+1 as one of the arguments:
TB s r
(ψ~r j+1, h j )− TB q (ψ~r j+1, h j )Δ t = 0.1218 m
[13]
The groundwater elevation is assumed unchanged in the intermediate solution step for the pressure heads; so h j is used. This
equation can be solved directly for ψr~j+1 by first constructing a
table for the total of the left side; the solution method is similar
to the one used by Veldhuizen et al. (1998) in a simplified surface
water model. The table can be interpreted as a water demand
function: the left side of Eq. [13] gives the total amount of water
needed for storage and percolation, in dependence on ψr~j+1.
The demand must be matched to the water availability, which is
0.1218 m in the example. As can be derived from Table 1 through
an inverse interpolation, a value of 0.1218 m for the availability matches pFr~j+1 = 1.62 [=1.80 + [(0.1218 − 0.1169)/(0.1222 −
0.1169)](1.60 − 1.80)], and thus ψr~j+1 = −0.41 m. This procedure
also yields sr~j+1 = 0.1180 m and qrave = qr~j+1 = −0.0038 m d−1.
www.vadosezonejournal.org · Vol. 7, No. 2, May 2008
F . 9. Numerical example for a me step of 1 d, star ng from an
equilibrium profile at t j (flux qj = 0). Indicated are the water balance terms for the “intermediate update” to the moisture profile
at me t~j+1, using the groundwater eleva on at t j. Explana on
of symbols: Iave is the me-averaged infiltra on rate (m d−1),
∆sr is the total volume change of the root zone (m), −qrave is the
me-averaged percola on rate from the root zone (m d−1), ∆ss is
the total volume change of the subsoil (m), and −qhave is the meaveraged groundwater recharge rate (m d−1).
775
hj+1,p and the newest total storage sj+1,p (total of the storage in the
control volumes of root zone and subsoil) do not form a point
on the storage curve. Figure 10 shows two options for updating
the phreatic storage coefficient, using either the level-based storage coefficient μh or the storage-based storage coefficient μs. The
first is the standard option. The second is used for handling the
nonlinear transition of the storage relationship at the soil surface
to avoid oscillations of h between iterations.
In the numerical example, the newest value of the total
saturated flow Gave,p is equal to 0.040 m d−1. The update of
the storage is done with a rearranged form of Eq. [14]: sj+1,p =
sr~j+1 + ss~j+1 + (−qhave + Gave,p)Δt = 0.1180 + 0.6702 + (0.0004
+ 0.0400)1 = 0.8286 m. This value of sj+1,p exceeds the saturated
storage by 0.0086 m; in that case, the groundwater model
will be given the storage-based storage coefficient for the next
solution cycle.
Having determined the (final) new groundwater elevation,
the pressure head of the root zone is finalized. In the numerical
example, the groundwater elevation rises to above the soil surface. In that case, the intermediate update of the pressure head
(ψr~j+1 = −0.41 m) is overruled by the groundwater update h.
Then the final value of the pressure head (ψrj+1) is adjusted to a
realistic value that conforms to the new groundwater elevation.
For further details on model implementation, see van Walsum
et al. (2006).
is from the intermediate update at t~j+1.The form of the water
balance given in Eq. [9] uses the root zone recharge R. Here the
procedure starts from t~j+1, so the flux −qhave to the phreatic
surface is used as the groundwater recharge. (The other part of
the recharge R has been used for the intermediate update.) The
water balance condition for the groundwater coupling can then
be written as
0
∫zs ⎡⎢⎣ θ
j +1
( z ) − θ ~ j +1 ( z )⎤⎥ d z = (−q have + G ave )Δ t
⎦
[14]
The integration of θ~j+1(z) is given by s~j+1. The integration of the
(unknown) θj+1(z) is replaced by a tabular storage function tbsg(h)
that is derived by summing the tabular functions tbsr(ψr,h) and
tbss(ψr,h) for the storage in the root zone and the subsoil; the
(known) intermediate update of the pressure head, ψr~j+1 is
inserted. The balance equation can now be written as
TB s g ( h )− s
~ j +1
= (−q have + G ave )Δ t
[15]
In principle, the groundwater model now has all the information
that is needed for solving the coupled flow formulation to solve
for the unknown h and G. But most groundwater models cannot
handle a nonlinear storage relationship. Therefore, an iteration
cycle was created in which the storage characteristics are passed to
the groundwater model in the form of a dynamically determined
phreatic storage coefficient. For the converged solution, the coefficient is given by
μ j +1 = (s j +1 − s ~ j +1 ) (h j +1 − h j )
Postprocessing
In the off-line postprocessing (after completing the online
simulations for the whole simulation period), detailed pressure
head and moisture content profiles are constructed using the
functions tbΨi(ψr,h) and tbΘi(ψr,h) that are available for the
compartments i of the schematization used by the Richards-type
model. The fluxes between the compartments are found from
repeated application of the water balance equation (cf. Eq. [10]):
[16]
where μ is the phreatic storage coefficient for the transition from
t~j+1 to t j+1 (m m−1). For the numerical example, the obtained
function tbsg(h) is given in Fig. 10. The function has been
extended above the soil surface, assuming a storage coefficient
of 1.0 for “ponded groundwater.” In the nonconverged situation
of the pth iteration, the newest groundwater elevation iteration
⎡ j +1 − θ j ) Δ t + τ ave ⎤ Δ z
q iave = q iave
+1 − ⎢⎣(θ i
i
i ⎥⎦
i
[17]
where qiave is the time-averaged flux density through the top of
compartment i (with the first one just below the soil surface)
during the increment from time t j to t j+1 (m d−1), Δzi is the
compartment thickness (m), and τiave is the time-averaged root
extraction rate (m3 m−3 d−1). The balances can be made by
starting from the top boundary condition and then proceeding
downward, or by starting from the bottom boundary condition
and then proceeding upward. If the start is made at the bottom
compartment N, the value of Gave (total of the time-averaged
saturated flow to the column) is used for qN+1ave. From there on,
Eq. [17] can be applied for i = N − 1, …, 1.
Plausibility Test
The modeling method is an approximate one compared with
models based on Richards’ equation. To obtain an impression of
its plausibility, a comparison was made with results obtained by
the SWAP model (Kroes and van Dam, 2003). Since a steadystate version of SWAP was used to derive the tabular functions,
we term our model MetaSWAP, where the prefix meta refers to
the aggregate scale of the online calculations.
A systematic test was performed using the 21 soil types
in the classification scheme for the Netherlands (Wösten et al.,
F . 10. Example of a tabular storage func on
sg(h) and its inverse
hg(s), showing two op ons for upda ng the storage coefficient
of the groundwater model. Explana on of symbols: hj is the
groundwater eleva on at me t j, hj+1,p is the pth itera on for the
groundwater eleva on at t j+1, s~j+1 is the “intermediate update”
of the total water storage at t~j+1, sj+1,p is the pth itera on for the
storage at t j+1, μhj+1,p+1 is the (p + 1)th itera on for the level-based
storage coefficient, and μsj+1,p+1 is the the (p + 1)th itera on for the
storage-based storage coefficient.
www.vadosezonejournal.org · Vol. 7, No. 2, May 2008
776
2. Physical proper es of soils according to the PAWN classifica on scheme for
1988; utilizing the Staring series of soil physical T
the
Netherlands
(Wösten et al., 1988). The parameter dc indicates the maximum depth
property types defined by means of Mualem
below the root zone for using Eq. [7] in the solu on procedure (Fig. 6). The values listed
parameters, revised in Wösten et al., 2001). here are rough first es mates.
These are listed in Table 2. Many regional
studies have used this nationwide classifica- Soil
Descrip on
dc
Area
no.
tion (Arnold and van Vuuren, 1988; Boogaard
m
%
and Kroes, 1998; Vermulst and de Lange, 1
Decomposed clayey peat over eutrophic peat: peat soil with decomposed
1.70 3.2
1999). For the verification, column models
topsoil
of 1 m2 were coupled to a one-layer “dummy” 2
Decomposed mesotrophic peat over a coarse-textured, sandy subsoil: peat
1.70 3.8
soil with decomposed topsoil
MODFLOW model with very low horizontal
Humic very fine textured, clay topsoil over eutrophic peat: peat soil with a
1.70 3.1
conductivity; effectively, this set the regional 3
clay cover
flux to zero. The drainage conductance was
4
Humic very fine textured, clay topsoil over coarse-textured, sandy subsoil:
1.70 0.7
set to 0.01 m2 d−1. Separate simulations were
peat soil with a clay cover
done for two drainage depths: 0.75 and 1.5 m. 5
Humic, medium-textured, sandy topsoil over coarse-textured, sandy
1.70 5.4
Grassland was used as the vegetation, with two
subsoil: peat soil with sand cover
Decomposed clayey peat over unripened clay: peat soil with decomposed
1.70 1.0
options for the root zone depth: 0.3 and 1.0 m. 6
topsoil
The models were run for weather conditions in
Aeolian, coarse-textured sandy soil: sandy soil
1.70 4.9
De Bilt (the Netherlands) in 1995 (wet spring, 7
8
Podzolic, coarse-textured sandy soil: sandy soil
1.70 1.3
dry summer, normal annual precipitation) and
9
Podzolic, medium-textured sandy soil: sandy soil
1.70 17.6
in 1976 (normal spring, extremely dry summer
10 Podzolic, medium-textured sandy soil over coarse-textured sand: sandy soil
1.70 1.4
occurring in <1% of years). MetaSWAP was 11 Podzolic, medium-textured sandy soil over boulder clay: sandy soil
1.70 4.3
run with a fixed time step of 1 d; SWAP was 12 Plaggen, coarse-textured sandy soil: sandy soil
1.70 5.8
run with a variable time step.
13 Humic gleysol, coarse-textured sandy soil: sandy soil
1.70 4.5
By doing trial runs with MetaSWAP, we 14 Podzolic, coarse-textured sandy soil: sandy soil
1.70 3.5
first (roughly) calibrated the maximum depth 15 Calcareous, medium-textured, clay soil: alluvial soil
2.10 11.0
2.10 11.1
under the root zone (dc), which acts as a crite- 16 Medium-textured clay soil: alluvial soil
1.70 5.9
rion for using Eq. [7] in the solution procedure 17 Fine-textured clay soil: alluvial soil
18 Fine-textured clay over mesotrophic peat: alluvial soil
1.70 3.4
(Fig. 6). Table 2 lists the values obtained.
1.70 5.7
To evaluate the performance of the 19 Medium-textured clay over sand: alluvial soil
20 Medium-textured clay over coarse-textured sand: alluvial soil
1.70 0.6
MetaSWAP model, the model efficiency coef21 Aeolian, medium-textured loam: loess soil
2.50 1.6
ficient (ME, dimensionless) as defined by Nash
and Sutcliffe (1970) was applied to the pre3. Dimensionless model efficiency coefficient (Nash and Sutcliffe, 1970)
T
dicted phreatic levels. Simulated values of the
of MetaSWAP, with groundwater eleva on simula ons by SWAP as a referMetaSWAP model were compared with results of the ence, for grassland on the 21 soil types from Table 2. The lateral groundwater
SWAP model. A value of 1.0 corresponds to a perfect flow was set to zero in the test.
fit. Also, the calculated actual evapotranspiration for the
Model efficiency coefficient
exceptionally dry year of 1976 was compared with that
Simula on year 1995
Simula on year 1976
obtained from SWAP; we did this by calculating the ratio
Root zone thickness, m
0.3
0.3
1.0
0.3
0.3
1.0
FE between the year totals. An FE value of 1.0 corre- Soil no.
Drainage depth, m
0.75
1.5
1.5
0.75
1.5
1.5
sponds to a perfect fit. From the given results in Tables 2, 1
0.97
0.90
0.76
0.86
0.90
0.79
3, and 4, it can be seen that
2
0.99
0.99
0.87
0.99
0.98
0.95
• for the year 1995, a root zone thickness of 0.3 m, and 3
0.95
0.68
0.82
0.83
0.88
0.80
a drainage depth of 0.75 m, 17 out of 21 soil types 4
0.96
0.99
0.91
0.94
0.95
0.95
have a groundwater ME that is >0.95 and cover about 5
0.99
0.99
0.80
0.99
0.99
0.95
85% of the area in the Netherlands;
6
0.90
0.51
0.30
0.69
0.69
0.42
• for the year 1995, a root zone thickness of 0.3 m, and 7
1.00
0.99
0.78
0.99
0.99
0.80
a drainage depth of 1.5 m, 14 out of 21 soil types 8
1.00
0.99
0.91
1.00
1.00
0.82
have a groundwater ME that is >0.95; these 14 cover 9
1.00
1.00
0.95
1.00
0.99
0.93
about 67% of the area; 16 out of 21 have an ME that
10
0.99
0.99
0.97
0.99
0.96
0.99
is >0.90 and cover about 75% of the area;
11
0.99
0.94
0.91
0.96
0.92
0.62
• for a root zone thickness of 1.0 m and a drainage depth 12
1.00
0.99
0.95
1.00
0.99
0.96
of 1.5 m, four out of 21 soil types have a groundwater
13
1.00
1.00
0.98
1.00
0.99
0.94
ME that is >0.95; 10 out of 21 have an ME that is
14
0.97
0.99
0.84
0.93
0.99
0.54
>0.90 and cover about 54% of the area;
15
0.99
0.97
0.94
0.96
0.91
0.91
• for the year 1976, the results show lower ME values,
16
0.97
0.85
0.79
0.92
0.88
0.82
but not dramatically lower; and
17
0.88
0.51
0.22
0.69
0.74
0.27
• the evapotranspiration totals simulated by MetaSWAP 18
0.92
0.85
0.80
0.91
0.97
0.67
are nearly always lower than those of SWAP, but
19
0.99
0.99
0.95
1.00
0.98
0.96
mostly this is only by a few percentage points; there
0.99
0.99
0.87
0.99
1.00
0.60
is one case (a clay soil covering 3% of the area) in 20
21
0.99
0.96
0.94
0.95
0.91
0.87
which the difference is 15%.
www.vadosezonejournal.org · Vol. 7, No. 2, May 2008
777
T
4. Comparison of evapotranspira on totals simulated for 1976 for grassland on the 21 soil types from Table 2. FE is the ra o between
the value obtained from MetaSWAP (ETMSWAP) and from SWAP (ETSWAP). The poten al evapotranspira on of grassland is 616 mm for the
simula on year.
Soil no.
Root zone thickness = 0.3 m,
drainage depth = 0.75 m
ETSWAP
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
ETMSWAP
————mm ————
530
505
592
574
492
469
500
472
558
543
466
452
509
498
541
533
576
568
588
581
577
576
567
559
592
586
412
402
556
555
515
511
405
397
425
415
572
569
554
547
501
489
Root zone thickness = 0.3 m,
drainage depth = 1.5 m
FE
0.95
0.97
0.95
0.94
0.97
0.97
0.98
0.98
0.99
0.99
1.00
0.99
0.99
0.98
1.00
0.99
0.98
0.98
1.00
0.99
0.98
ETSWAP
ETMSWAP
————mm ————
488
490
549
529
459
467
464
448
510
498
449
453
397
389
429
423
494
483
492
480
501
487
501
491
510
502
336
345
527
528
487
491
390
406
400
410
522
517
453
453
479
469
The test results provide only a first indication of the applicability of the simplified modeling concept. If the criterion for the
ME is set at 0.90, then the tests roughly indicate that for thin root
zones (<0.5 m thick) and shallow groundwater elevations (within
2 m of the soil surface), the model performs satisfactorily in about
75% of the area. More tests will follow, using simulations that
are coupled to a realistic groundwater model instead of a dummy
model with a zero regional flux, as was used in the current test.
Due to the influence of feedback mechanisms in the coupled
system, we expect that the modeling efficiencies will increase.
FE
1.00
0.96
1.02
0.96
0.98
1.01
0.98
0.99
0.98
0.98
0.97
0.98
0.98
1.03
1.00
1.01
1.04
1.02
0.99
1.00
0.98
ETSWAP
ETMSWAP
————mm ————
615
584
616
608
613
570
615
562
616
594
552
521
552
482
558
506
599
553
543
526
557
519
614
576
596
560
422
385
610
598
597
566
493
452
600
510
608
584
507
476
579
546
FE
0.95
0.99
0.93
0.91
0.96
0.95
0.87
0.91
0.92
0.97
0.93
0.94
0.94
0.91
0.98
0.95
0.92
0.85
0.96
0.94
0.94
comparison between the results of our approach and that of De
Laat has yet been performed.
For deep groundwater elevations and for situations with percolation (switch in flow chart of Fig. 6), the method presented
does not under all circumstances assume preservation of pressure head continuity at the interface between the root zone and
the subsoil. The preservation assumption seems to be an obvious choice, because it follows from “sound physics.” Inevitably,
however, a method that uses (segments of ) steady-state profiles
at some point makes concessions with respect to the reality of
unsteady flow governed by Richards’ equation. The abrupt transitions in moisture profiles in our schematization method are
needed to avoid nonsensical simulation results where groundwater elevations are deep. We also prefer to use composite profiles for
situations with percolation, because we think they better simulate the dynamics following a heavy precipitation event. This is
elaborated below.
Figures 11 and 12 depict the moisture and flux profiles simulated by SWAP and MetaSWAP for the example above under
model implementation; here the groundwater elevation is held
constant. After a major precipitation event of 0.016 m, there are
large differences between the two models. MetaSWAP simulates
a deeper penetration of water content changes during the time
step. There are three reasons for this:
• the assumed instantaneous spreading of infiltration water
throughout the root zone,
• the use of steady-state water content profiles in the solution
procedure, and
• the use of a fully implicit scheme for the time weighting of the
moisture flux (f = 1 in Eq. [5] and [6]).
Discussion
For situations where the soil is drying out and groundwater elevations are shallow, we assume that the steady-state flux
is active at the phreatic surface. For a drying soil, the volume
change rate is negative; thus it follows from Eq. [10] that the
calculated capillary rise at the root zone boundary (qrave) is greater
than the flux density at the phreatic surface (qhave). The latter is
determined from one of the profiles in the database of steady
states. The justification for simulating the higher-than-steadystate flux density through the root zone boundary is the presence
of unsteady gradients that in reality exist due to the root-zonedriven drying process of the subsoil. For deep groundwater levels,
this method yields unrealistically high values of qrave. That is
avoided by switching to the alternative solution method (flow
chart in Fig. 6), which uses a composite profile as in Fig. 3. The
method presented by De Laat (1976) places the flux density of a
steady-state profile at the root zone boundary. Also, De Laat does
not use “composite” profiles in the way we do here. No direct
www.vadosezonejournal.org · Vol. 7, No. 2, May 2008
Root zone thickness = 1.0 m,
drainage depth = 1.5 m
778
F . 12. Comparison with the SWAP model (Kroes and van Dam,
F . 11. Comparison with the SWAP model (Kroes and van Dam,
2003), displaying the flux density profiles for the numerical example
given in Fig. 9. Shown are the me-averaged values for a simula on
period of 1 d; Iave is the me-averaged infiltra on rate (m d−1).
2003), displaying the water content profiles for the numerical example given in Fig. 9; Iave is the me-averaged infiltra on rate (m d−1).
To start with the last point, the accuracy of the model would
have few consequences. For instance, in most water quality
(in this example) have benefited from the use of a semi-implicit
simulations, it is more important to simulate the water balance
scheme for the time weighting of the moisture flux by setting f
accurately at a larger time scale than 1 d. The percolation rate and
= 0.5 in Eq. [5] and [6]. The semi-implicit scheme yields a timegroundwater elevation at a time scale of 10 to 30 d determines
averaged flux density through the root zone boundary of 0.0026
the removal of substances and the influence of aeration on biom d−1, which (by some degree of coincidence) is nearly exactly
chemical processes. And the occurrence of saturation-generated
the same as the value of 0.0027 m d−1 that is simulated by the
runoff due to a major precipitation event is determined by the
SWAP model. But even if the simulated flux density at the root
water balance simulation of the preceding period, not by the
zone boundary is correct, there will still be substantial deviations
exact dynamics of soil moisture. In situations where the model
in the subsoil. Owing to the use of steady states, MetaSWAP
deficiency does indeed play a role, knowledge about it can be used
transmits effects on the flux to the groundwater sooner than
for filtering the results. In doing inverse modeling, for instance,
SWAP does. By breaking the profile into two parts, this disgroundwater elevation observations directly following a major
advantage of the quasi-steady-state modeling method has been
precipitation event can be left out of the analysis.
made smaller. The modeling accuracy could be increased further
by using more than one profile
segment for the subsoil.
To gain further insight
into the differences between
the model outcomes, the simulated recharge, flux through
the bottom of the root zone,
and groundwater elevation are
presented in Fig. 13 for the
simulation year 1995. The simulated day-averaged recharge rates
are nearly identical. The plots
of the day-averaged flux density
through the bottom of the root
zone reveal that SWAP simulates
higher percolation pulses after
major precipitation events. The
SWAP model has more wave-like
dynamics, which would become
even more apparent if “momentaneous” flux densities would
have been compared with the
per-day time-averaged values of F . 13. Time series comparison with the SWAP model (Kroes and van Dam, 2003), showing the day-averMetaSWAP. In many practical aged recharge rate R (m d−1), the simulated day-averaged moisture flux density through the bo om of
applications, this conceptual the root zone qr, taken posi vely upward (m d−1), and the groundwater eleva on h (m) (The simula on is
limitation of MetaSWAP will for Soil Type 21 of Table 2, using the simula on year 1995, De Bilt, the Netherlands.)
www.vadosezonejournal.org · Vol. 7, No. 2, May 2008
779
The capillary rise simulation of MetaSWAP (Fig. 13) shows
erratic behavior at some points; for instance, there is a spike at the
end of July. This artifact is due to the transition from percolation
to capillary rise, when the root zone has already dried out beyond
the equilibrium profile while the subsoil is still percolating. Any
deficit with respect to the equilibrium moisture content of the
root zone is then directly supplied from the subsoil, so the model
then simulates a capillary rise flux density that is equal to the total
root water uptake.
The simulated groundwater elevation of the SWAP model
(Fig. 13) reaches a lower level than that of MetaSWAP during
the summer. This is due to differing dynamic effects within the
soil profile: As in the numerical example (see Fig. 11 and 12),
MetaSWAP is quicker to transmit the effect of changing conditions in the root zone to the phreatic surface. In this case, the
change involves the onset of less dry conditions.
Given the possibility of doing high-resolution modeling with
the method, there is no need to combine it with the upscaling (on
the horizontal plane) of soil physical properties. Rather, there is
a need to downscale the available information. In this context, it
is relevant to mention that the physical properties of the 21 soil
types listed in Table 2 were obtained using a form of averaging
among the available samples (Wösten et al., 2001). There is a
general feeling that the data give too optimistic results for the
simulated capillary rise flux. For the exceptionally dry year of
1976 (with a recurrence period of 100 yr), the simulated evapotranspiration totals do not show the expected depression. This
is partly due to the simulations’ lack of feedback to vegetative
development. Nonetheless, new surveys—and reinterpretation of
existing information—are expected to reveal that the main part
of the problem is in the soil physical data.
The model is still being improved. For example, we are considering using the pressure head at the top of the profile as an
independent variable in the tabular functions (as is done in the
description of the method) instead of the mean pressure head in
the root zone. Possibly this could improve the simulation of soil
columns with thick root zones. Another option for improvement
is to use two profile segments for the subsoil below the root zone.
This could improve the simulation of situations where groundwater elevations are deep, as already mentioned with respect to the
simulation of percolation pulses. A second segment also makes it
possible to simulate capillary rise of moisture from the top part
of the subsoil. (In the current method, the capillary rise is near
zero for deep groundwater elevations.) This can be effected by, in
situations with an upward flux, letting the profile segments of the
root zone and the top part of the subsoil fit together.
from “lumped” models. The results can subsequently be used for
water quality modeling and for agroecological evaluations. To
date, no case studies using the method have yet been documented.
Nonetheless, its operationality has been tested on regional scale
(>40,000 ha) for a 25- by 25-m2 resolution, using a single standard PC (Veldhuizen et al., 2006), involving more than 700,000
soil-column models in combination with a MODFLOW groundwater model (Harbaugh et al., 2000) and a simplified surface
water model.
The method can be termed an experimental approach, because
it contains steps that are partly based on intuitive notions. An
example is the used “partitioning” of the moisture content change
to the next time level. This partitioning enables a straightforward
solution procedure involving steps that are performed consecutively. Due to its experimental nature, the method requires testing
for its justification. To that end, a plausibility test was performed
using results of a Richards-type model as a reference. The tests
roughly indicate that for thin root zones (<0.5 m thick) and shallow groundwater levels (within 2 m of the soil surface), the model
performs satisfactorily in about 75% of the area. We expect that
similar results could be obtained for other lowland regions. The
tests were performed without a coupling to a dynamic groundwater model. In a coupled system, the differences between the
quasi-steady simulations and the ones based on Richards’ equation can be expected to diminish due to feedback mechanisms
between the groundwater and the unsaturated zone. Therefore,
we expect that new tests will indicate higher model accuracy.
For the remaining soils that were not simulated satisfactorily, a
more advanced modeling method will have to be used. Applying
the simplified approach wherever possible enables the available
computational resources to be used efficiently. This creates more
room for doing sensitivity analyses and determining parameters
through inverse modeling. This could help improve model reliability and—perhaps more importantly—be instrumental for
quantification of the reliability.
A
In the course of developing and testing this method, we have had
many stimulating discussions with colleagues Ab Veldhuizen, Pim Dik,
Joris Schaap, Frank van der Bolt, Jan van Bakel, Ger de Rooij, and Paul
Torfs. Joris Schaap and Pim Dik provided the test results for the soil
schematization of the Netherlands.
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