Introduction
The use of vegetative filter strips (VFSs) can reduce sediment and surface
runoff (RO) pollutant (i.e., sediment, colloids, nutrients, pesticides, pathogens)
movement into receiving water bodies. The dense vegetation–soil system
reduces runoff pollutants in three ways by increasing (a) soil infiltration
that reduces total runoff volume (and dissolved runoff pollutants),
(b) surface roughness that reduces surface velocity and produces settling of
sediment and sediment-bonded pollutants, and (c) contact between dissolved and
particulate pollutants with the soil and vegetation surfaces that enhances
their removal from runoff (Muscutt et al., 1993; Muñoz-Carpena et al.,
1999, 2010; Dosskey, 2001; Fox et al., 2010; Yu et al., 2013; Lambretchs et
al., 2014; Wu et al., 2014). The efficiency of VFS in trapping pollutants is
heavily influenced by the highly variable spatial and temporal dynamics
introduced by site-specific combinations of soil, climate, vegetation, and
human land use. For the case of runoff pesticides, these influences have been
recognised in multiple field studies (Lacas et al., 2005; Reichenberger et
al., 2007; Poletika et al., 2009; Sabbagh et al., 2009). Other effects like
hydraulic loading under concentrated flow conditions (Fox et al., 2010) or
timing of the pesticide application (Sabbagh et al., 2013) can also result in
reduced filter trapping efficiencies. As these systems are complex, the
practice of using generic, simple regression equations relating the reduction
efficiency of pollutants with VFS physical characteristics (i.e., length,
slope) is often inadequate (Fox and Sabbagh, 2009).
Mechanistic understanding of VFS behavior has advanced significantly in the
last 20 years, and numerical simulation tools are available to analyze this
important best management practice (BMP) under upland field conditions where
runoff is governed by excess rainfall and field inflow processes
(Muñoz-Carpena et al., 1993, 1999; Abu-Zreig, 2001; Muñoz-Carpena and
Parsons, 2004; Poletika et al., 2009; Sabbagh et al., 2009; Carluer et al.,
2017). A recent linked mechanistic model has investigated multiple input
factors and their relative importance and uncertainties on the predicted
reduction of runoff, sediments, and pesticides (Fox et al., 2010; Lambrechts
et al., 2014; Muñoz-Carpena et al., 2010, 2015).
However, because of their location near or at the riparian zone, VFS can at
times be bounded by a seasonal shallow water table (WT) (Borin et al., 2004;
Ohliger and Schulz, 2010). Examples of areas where these
conditions exist either seasonally or on a more permanent basis are humid
coastal flatland zones, floodplains near water bodies, and soils with
limiting horizons resulting in perched WTs. Generally, capillary effects from
a WT can reduce infiltration and increase subsequent runoff processes as well
as
have a major effect on contaminant transport to surface waters (Gillham,
1984). In spite of the potentially important environmental impacts of the
presence of shallow water under VFS, there is a dearth of studies addressing
this problem either experimentally or mechanistically. Several authors
suggest the importance of this factor in VFS experimental studies (Lacas et
al., 2005; Arora et al., 2010) or when designing or implementing this field
BMP (Simpkins et al., 2002; Dosskey et al., 2006, 2011), but they do not
provide a mechanistic interpretation. Some authors suggest that the reduction
of infiltration and VFS efficiency can be problematic for seasonal WT depths
above 2 m, typical of hydromorphic soils (Dosskey et al., 2006, 2011; Lacas
et al., 2012). As cited by Salvucci and Entekhabi (1995), the importance of
accounting for areas of WT effects in water balance and runoff studies has
been recognized for a long time, and specialized analysis and simulation
approaches have been proposed by numerous authors (for example, Vachaud et
al., 1974; Srivastava and Yeh, 1991; Salvucci and Entekhabi, 1995; Chu, 1997;
Basha, 2000).
In spite of the ubiquity and importance of these areas and previous
specialized analysis and modeling efforts, commonly used field and watershed
hydrological models are limited when describing infiltration and soil water
redistribution with WT (Beven, 1997, Liu et al., 2011). Among existing
simulation approaches, solutions to the fundamental Richards (1931) partial
differential equation (RE) can describe the infiltration and redistribution
of water in soil, including the specific case of when a system contains a WT.
However, RE does not have a general analytical solution and its application
in real-world systems requires computationally intensive numerical
approximations that can result in mass-balance and instability errors in some
cases (e.g., for coarse soils and highly dynamic boundary conditions) (Celia
et al., 1990; Paniconi and Putti, 1994; Miller et al., 1998; Vogel et al.,
2001; Ross, 2003; Seibert et al., 2003). As a result, soil infiltration is
often modeled in field and watershed models using simpler physically-based
approaches (Jury et al., 1991; Smith et al., 1993; Haan et al., 1994; Singh
and Woolhiser, 2002; Talbot and Ogden, 2008; Ogden et al., 2015). One of the
most often used approaches in hydrologic modeling is the Green–Ampt (1911)
model adjusted to account for variable rainfall (Mein and Larson, 1973; Chu,
1978; Skaggs and Khaleel, 1982). The model has the advantages of being
computationally efficient and that its parameters can be directly estimated
from physical measurements or derived indirectly from soil texture (Rawls et
al., 1982, 1983). However, the limitation of the original Green–Ampt model
is that it assumes isotropic soil with uniform initial moisture content and
saturated “piston” infiltration. Even with these nonrealistic assumptions,
if effectively parameterized, this method still generates useful and reliable
results compared with other numerical and approximated approaches (Skaggs et
al., 1969; Mein and Larson, 1973). Considering its advantages, Bouwer (1969)
highlighted the utility of this method when taking into account the
computational trade-offs with RE solutions.
Conceptual depiction of infiltration and soil water redistribution
for soils with shallow water table for (a) time before wetting front reaches
the water table and (b) time after the wetting front reaches the water table
(t≥tw), where surface infiltration flow (Qf) is limited by
lateral Boussinesq subsurface flow (QL). See explanation of symbols in
the text.
Conceptual curves of (a) infiltration rate, f; (b) cumulative
infiltration, F; and (c) soil water redistribution, θ, under shallow
water table, for soil without initial ponding, and constant rainfall rate
(i) conditions. The singular times for ponding (tp), shifting (t0)
and to reach column saturation (tw), and final infiltration rate
(fw) after the wetting front reaches the water table (t≥tw)
are represented.
Extensions of the Green–Ampt model beyond its initial assumptions have
enabled its application to other natural infiltration cases, such as
nonuniform soil profiles (Bouwer, 1969; Beven, 1984) and multistorm
infiltration and redistribution (Ogden and Saghafian, 1997; Smith et al.
2002; Gowdish and Muñoz-Carpena, 2009). A particularly important case
where an extension of the original assumption of the Green–Ampt model is
necessary is when there is a WT. In general, depth-averaged soil moisture
values in traditional infiltration equations like Green–Ampt (i.e.,
semi-infinite, uniform initial soil moisture) overpredict infiltration
estimations when the soil is bounded by a WT. This is due to the difficulty
in obtaining an equivalent initial uniform soil water content that
effectively represents the real nonuniform water content condition with WT
(Salvucci and Entekhabi, 1995; Chu, 1997). Recently, Liu et al. (2011)
presented a modification to Craig et al.'s (2010) nondimensional form of the
Green–Ampt model to account for the presence of a WT. Although this
modification is shown to provide acceptable results compared with a RE
solution for a range of WT depths, the method assumes an initial uniform soil
water content profile, and its performance relies on an empirical correction
between RE and standard Green–Ampt results. Alternatively, previous works
(Childs, 1960; Holmes and Colville, 1970; Duke, 1972) have suggested
describing the soil water redistribution over a WT as an equilibrium
hydrostatic condition (Fig. 1). This approach assumes a linear relationship
of soil matric potential (h, [L]) and soil depth (z, [L], positive downwards from the surface),
whereby the nonuniform water content of the soil (θ,
L3 L-3) is described by the soil water characteristic curve,
θ=θ(h) (Jury et al., 1991):
h=L-z⇒θ=θ(L-z),
where L [L] is the depth of the fixed shallow water table below the soil
surface (i.e., the distance from the surface). Based on these initial and
boundary hydrostatic equilibrium conditions, Chu (1997) proposed an
incremental calculation technique to evaluate infiltration into ponded soils
with a WT. This calculation relies on Bouwer's (1969) expression of the
Green–Ampt equation that accounts for infiltration of water into a
nonuniform soil as follows:
t=∫0zF1fθs-θ(L-z)dz,
where t [T] is time since the beginning of the event, θs
[L3 L-3] is the saturation water content, f [LT-1] is the
rate of surface infiltration, and zF [L] is the wetting front depth.
Following Neuman (1976) and Chu (1997), after substitution of the equilibrium
condition (Eq. 1), the cumulative (F, [L]) and instantaneous infiltration
(f, [LT-1]) can be calculated as follows:
F=Fp=∫0zF[θs-θ(L-z)]dz=θszF+∫LL-zFθhdh=θszF-∫L-zFLθ(h)dh,f=fp=Ks+1zF∫0L-zFK(h)dh=Ks+-1zF∫LzFK(L-z)dz,
where the subscript “p” denotes under-ponding or “capacity”, i.e., when the
flux at the surface is not limited by available water and is therefore
maximum for each time; Ks and K(h) [LT-1] represents the
soil saturated and unsaturated hydraulic conductivity, respectively.
Chu (1997) proposed the solution to Eqs. (2)–(4) using sufficiently small
increments of z, Δz=zi-zi-1. If an initial value of F1
and f1 for the first Δz (from the surface to a small depth) is
known, then successive values of time (ti=ti-1+Δt) for each
Δz can be approximated by substituting Eq. (3) into Eq. (2) as
follows:
ti=ti-1+dt=ti-1+Fi-Fi-10.5(fi+fi-1).
Chu (1997) further proposed that a valid initial step could be obtained by
assuming standard Green–Ampt conditions (i.e., piston flow) from the surface,
hydrostatic equilibrium of the surface water content with the WT (θo), and calculating the suction at the wetting front
(Sav) as follows (Bouwer, 1964):
Sav=1Ks∫0LK(h)dh.
Vachaud et al. (1974) was able to use experimental data to test the solution
of this equation successfully. However, their experimental data did not allow
enough time to determine how the model would respond when the wetting front
reaches L.
An elegant and useful approximate solution to ponded infiltration with the WT was
proposed by Salvucci and Entekhabi (1995). Their solution is based on the
assumptions of initial hydrostatic equilibrium and uses Philip (1957)
integral approximation of RE (Fig. 1). This approximate solution is
advantageous, as it describes not only the infiltration but also soil water
redistribution during infiltration, and the characteristics of the wetting
front as it moves towards the WT during long events. In addition, the method
assumes a more realistic piecewise linear wetting front with a variable slope
during infiltration (α in Fig. 1). This algorithm was successful when
compared with RE solution for three different soil types and when tested with
the soil moisture profile data from Vachaud and Thony's (1971) experiments.
However, the applicability of the algorithm for coupling with commonly used
hydrological models is limited as it requires ponded conditions, Brooks and
Corey's soil water function (Appendix Eq. A1), and similarly to the original
Green–Ampt it requires an implicit solution.
The overall objective of this work and its companion paper (Lauvernet and
Muñoz-Carpena, 2018) is to analyze the impact of the presence of a WT on
VFS efficiency. In this first paper, we will expand the Green–Ampt-based
infiltration solution to soils bounded by a WT under variable rainfall with
no initial ponding. We accomplish this by combining Salvucci and
Entekhabi (1995) and Chu (1997) approaches with a generic solution technique
and developing novel integral formulae to calculate the singular times (time
to ponding, tp, shift time, t0, and time to column
saturation, tw) for soils with no initial ponding. We assess the
ability of the simplified method to accurately predict surface infiltration
and water content predictions for a variety of soils compared with RE
numerical solutions and previously published experimental data. An
illustrative example of calculation during an unsteady rainfall event is also
presented along with examples of applications of the proposed algorithm to
analyze the effects of WT depth. In the companion paper, we couple the new
shallow water infiltration algorithm with an existing VFS numerical model
(VFSMOD, Muñoz-Carpena et al., 1999, 2010, 2015) and analyze the effects
on runoff, sediment, and pesticide removal efficiency.
Proposed algorithm
Infiltration rate in soils bounded by a WT with a nonponded
initial state and subject to constant rainfall
In general, the infiltration rate (f, LT-1) of a WT-bounded soil with
uniform rainfall rate (i, LT-1) and no initial surface ponding will
have a similar profile to the example shown in Fig. 2a, described by the
following:
f=i0<t≤tpf=fptp<t<twf=minfw,it≥tw
The identification of three singular times during the infiltration
calculations is necessary for a solution to Eq. (7). These singular times
are time to reach ponding (tp) (which depends on the shift time,
t0, described later) and time to column saturation (tw),
when the wetting front approaches the WT capillary fringe at depth
zw (see Fig. 1). The effective saturation depth zw
relies on L and soil air entry pressure (hb),
zw=L-hb(zw≥0; zw=0 when
L < hb, i.e., the soil is effectively saturated by the
capillary fringe). Often, hb is set at 0 (i.e., zw=L),
even if some of the soil characteristic functions take the air entry pressure
into account (Brooks and Corey, 1964; Clapp and Hornberger, 1978). At
tw, the soil column is saturated and the rate of infiltration
sharply drops to fw, or i if i < fw
(Fig. 2a). The value of tw depends on L and the slope of K(h) (Salvucci
and Entekhabi, 1995). If the WT is very shallow, the time to saturation
tw can occur before the time to ponding. Salvucci and
Enthekabi (1995) and Liu et al. (2011) initially proposed that the
infiltration rate is equal to fw=Ks, when t≥tw, meaning that the vertical hydraulic gradient at the initial WT
is 1. However, in most field situations when the wetting front has reached
the WT, the profile's hydraulic gradient is less than 1 and the proposed
solution might overestimate the final infiltration rate. Instead, another
solution is to consider that for t≥ tw the infiltration
flow at the surface (Qf) is controlled by lateral drainage flow
(QL) at the downslope boundary of the simulated soil elementary
volume (Fig. 1b), applicable to floodplain conditions typical of VFS. If we
consider that the soil profile is saturated at t≥tw, with an effective saturation depth zw=L-hb, following
Dupuit–Forchheimer assumptions (Van Hoorn and Van Der Molen, 1973) the
discharge (Qf=QL) can be estimated as follows:
Qf=fwwbQL=KshSwzwQf=QL⇒fw≈KshSozwb,
where Ksh is the lateral (horizontal) soil saturated hydraulic
conductivity, w and b are the width (VFS dimension perpendicular to the
flow) and length (VFS dimension in the flow direction) of the VFS surface
area, and S is the slope of the initial WT. In hillslope hydrological
modeling S is typically assumed to equal soil surface slope
(So) (Beven and Kirkby, 1979; Vertessy et al., 1993). If the
position of the infiltration elementary volume is close to a draining stream
where S>So, Eq. (8) may underestimate the
infiltration rate and a 2-dimensional drainage approach like Hooghoudt (1940)
equation should be used instead (Kao et al., 2001; Ritzema, 1994; van
Schilfgaarde, 1957). In the algorithm developed here, the two options for the
boundary condition are implemented: with “lateral drainage” (Eq. 8) and
Vachaud's “vertical drainage” (fw=Ks). In some field
conditions, a mixture of both end-time boundary conditions might be expected,
requiring empirical weighing of the two conditions.
Calculation of singular time points
Following Mein and Larson (1973), time to ponding tp is the time
for fp=i (intersection of the curves in Fig. 2a), typically
when the surface water content is equal to saturation (Fig. 2c). At t=tp the equivalent wetting front depth (zp) can be
calculated by equating Eqs. (4) and (7):
fp=ifp=Ks-1zp∫LzpK(L-z)dz⇒zp=-1i-Ks∫LzpKL-zdz.
Since Eq. (9) is implicit in zp≥0, it can be solved for each
time step by defining the function Gp: R→R and its
derivative Gp′ so that the root zp∈[0,
zw] (i.e., Gp (zp)= 0) is the wetting
front depth at tp,
Gpzp=zp+1i-Ks∫LzpK(L-z)dzGp′zp=1+1i-KsK(L-zp)
where zp can be obtained applying a bracketed Newton–Raphson
algorithm (Press et al., 1992) obtaining the following:
zpk+1=zpk-Gp(zpk)Gp′(zpk)withzpk+1-zpk|<ε,
where k is the Newton–Raphson iteration level, and ε the error
tolerance (here ε= 10-8). From Eq. (3) at t=tp
and z=zp, and Fp=i⋅tp we obtain the
following:
tp=1i(θszp-∫0zpθ(L-z)dz),
Next, to ensure that Fp (Eq. 3) and F=i⋅tp match
at the intersection of the two curves on t=tp (Fig. 2b), an
abscissa translation (shift time, t0) is applied to
Fp (Mein and Larson, 1973). Setting z=zp on Eqs. (2)
and (3) yields t0 as follows:
t0=∫0zp1fpθs-θ(L-z)dz.
Lastly, tw is determined by calculating the integral Eq. (2) at
zF=zw=L-hb (Fig. 1) and adjusting for
tp and to,
tw=tp-t0+∫0zw1fpθs-θ(L-z)dz,
and using Eq. (3), the cumulative infiltration at tw is
determined by the following:
Fw=θszw-∫hbLθ(h)dh.
The value of tw is equivalent to the nondimensional time Xc
proposed by Liu et al. (2011) that relies on the empirical error correction
between RE solution and the Green–Ampt model. However, here tw
(Eq. 14) is calculated analytically for the more general case of nonuniform
soil water content.
Infiltration capacity algorithm after surface ponding
The solution of Salvucci and Entekhabi (1995) can be simplified by setting
the wetting front slope to zero (i.e., a horizontal front (α=0) at the
depth zF, Fig. 1). This approach reduces the solution, making it
analogous to Eq. (2), which was employed by Bouwer (1969) in his explanation
of the Green–Ampt model's applicability. For initial nonponding conditions,
the equation becomes
t=tp-t0+∫0zFθs-θ(L-z)Ks-1zF∫LzFK(L-z)dzdzfortp<t<tw.
As the wetting front travels deeper into the soil, α could increase,
contingent on the type of soil (e.g., α is larger for fine soils).
However, as the wetting front approaches WT, the pore space available for
infiltration is small, which limits the error of the calculations (Salvucci
and Entekhabi, 1995). This assumption is tested in Sect. 2.4.
For a given time t, to solve for zF we specify the implicit
function of Eq. (16) G: R→R and its derivative G′, so that the root
zF∈[zi-1, zw] of the function G is equal
to the new depth of the wetting front,
G(zF)=t-tp+t0-∫0zFθs-θ(L-z)Ks-1zF∫LzFK(L-z)dzdzG′(zF)=-θs-θ(L-z)Ks-1zF∫LzFK(L-z)dz⇒zFk+1=zFk-G(zFk)G′(zFk)with|zFk+1-zFk|<ε.
In summary, for each time increment the proposed algorithm computes the depth
of the wetting front (Eq. 17), F (Eqs. 3, 15) and f (Eqs. 4, 7 and 8)
using the singular times auxiliary Eqs. (12)–(14). A bracketing step in the
Newton–Raphson algorithm is necessary, as the function G is undefined
outside its physical range
(zp < zF < zw) (Press
et al., 1992). The proposed algorithm is generic in that it can be used with
any soil hydraulic functions like those of Gardner (1958), van
Genuchten (1980) or Brooks and Corey (1964) (Appendix A) if numerical
integration is used. Here, we used a Gaussian-quadrature integration scheme
(Abramowitz and Stegun, 1972; Press et al., 1992).
Infiltration of soils with a WT and variable rainfall without initial
ponding
For real VFS field situations, unsteady rainfall without initial soil ponding
must be considered. Nonuniform rainfall is described by a hyetograph as a
series of constant rainfall periods j (i.e., i=ij for
tj < t < tj+1). The runoff produced by excess
infiltration (i.e., Hortonian) and WT saturation (i.e., Dunne) are then
determined at each time by water balance at the surface without accounting
for evaporation during the rain event (Chu, 1997):
ΔP=ΔF+Δs+ΔRO⇒ΔP-ΔF=Δs+ΔRO,
where Δ is the increment for that rainfall period, P and RO [L]
are cumulative precipitation and runoff (excess rainfall), respectively, and
when present s is the surface storage
(0 < s < smax) that acts as a reservoir
that must be filled (s=smax) before runoff is generated (Chu,
1978; Skaggs and Khaleel, 1982). For each period, if there is excess at the
surface (ΔP-ΔF > 0), the excess is first
distributed to fill up the surface storage (Δs ≤ smax-s) and the remainder (if any) to runoff (ΔRO =ΔP-ΔF-Δs≥0).
For nonponding conditions at the beginning of the event, tp and
t0 must be calculated (Eqs. 12–14), otherwise if initial ponding is
present tp=t0=0. If during a rainfall period the surface
storage becomes zero, and if the new ij+1 of the following period is larger
than the infiltration rate at the end of the last period, tp (and
t0) must be recomputed (Eqs. 12–14) for the subsequent rainfall event
(Chu, 1978; Skaggs and Khaleel, 1982). Also, each time tp, and
t0 are calculated, tw has to be recalculated.
To allow for predictions of soil water content redistribution during the
event (Fig. 1) and to maintain mass balance during infiltration for
alternating periods of ponding and nonponding conditions, it is necessary to
track the “effective” position of the wetting front zF for
periods with no ponding. To do this, the value of zF must satisfy
the total cumulative infiltration amount at every time step (Fig. 1) and F
(Eq. 3) becomes implicit in zF. As before, the root
zF∈[zj-i, zw] (zj-1 is the wetting
front depth at the previous time step) of the function GF:R→R
and its derivative is as follows:
GF(zF)=F-∫0zF[θs-θ(L-z)]dzGF′(zF)=-θs+θ(L-zF)⇒zFk+1=zFk-GF(zFk)GF′(zFk)with|zFk+1-zFk|<ε.
The wetting front depth estimates provided by the algorithm are key in many
hydrological applications where the aim is to simulate the potential for
direct contamination of the WT by pollutants.
Parameters used in numerical and experimental testing of SWINGO.
Nash–Sutcliffe coefficient of efficiency (Ceff) and root mean error
(RMSE) represent SWINGO infiltration goodness of fit with Richards' finite-difference solution (CHEMFLO-2000) and experimental data from Vachaud et
al. (1974).
Numerical testing1
Soil
L
θr
θs
Ks
hb
λ
η
Ceff3,4
RMSE3
(m)
(m s-1)
(m)
f(×10-6 m s-1)
Silty loam
1.5
0
0.35
3.40 × 10-6
0.450
1.20
4.67
0.994 [0.969–0.999]***
0.309 [0.119–0.541]
Clay
1.5
0
0.45
3.40 × 10-7
0.900
0.44
7.54
0.999 [0.998–1.000]***
0.015 [0.003–0.028]
Sandy loam
1.5
0
0.25
3.40 × 10-5
0.250
3.30
3.61
0.985 [0.882–0.998]***
1.326 [0.578–2.414]
Vachaud and Thony (1971)
1.01
0
0.35
1.75 × 10-5
0.181
0.73
4.63
0.927 [0.821–0.977]***
1.488 [0.893–2.240]
Experimental testing1,2
Soil
L
θr
θr
Ks
αvG
n
m
αGrd
nGrd
Ceff3,4
RMSE3
Vachaud et al. (1974)
(m)
(m s-1)
(m-1)
(m-1)
f(× 10-6 m s-1)
vG : vG
0.925
0.107
0.34
2.64 × 10-5
1.143
2.363
0.652
–
–
0.913 [0.742–0.951]**
6.204 [4.726–7.920]
vG : Grd
0.925
0.107
0.34
2.64 × 10-5
1.143
2.363
0.652
0.136
2.151
0.942 [0.828–0.971]***
5.069 [3.696–6.468]
1 hb, λ, η are the Brooks and Corey
parameters.
2 αvG, n, and m are van Genuchten parameters, and αGrd and nGrd are the Gardner parameters (see Appendix for
details). 3 median value (95 % confidence interval in
brackets). 4 models were statistically acceptable at
a significance level of 0.01 (***) or 0.05 (**) (Ritter and Muñoz-Carpena, 2013).
The next section provides an illustrative application of the full algorithm
(herein referred to as SWINGO: Shallow Water table INfiltration alGOrithm)
under unsteady rainfall conditions, typical in VFS settings (see the Supplement for coding details, source code, inputs, and outputs).
Testing and applications
Numerical testing
A first step to validate SWINGO is done for the case of initially ponded soil
and steady rainfall by a comparison with a finite-difference
mass-conservative numerical solution of RE (Celia et al., 1990) using
Nofziger and Wu's (2003) CHEMFLO-2000 model. We used four soils that
represented a variety of attributes. The Brooks and Corey soil water
attributes and hydraulic conductivity curves (Table 1) were used for the
initial soil description, and this description was later compared with van
Genuchten parameters yielding similar results (results not shown). The
first three soils represent typical clay, silty loam, and sandy loam soils with a
1.50 m deep WT (Salvucci and Entekhabi, 1995). The fourth soil corresponds
to a fine sandy soil experimentally studied by Vachaud and Thony (1971) with
a WT at 1.01 m.
Comparison of normalized infiltration rates (f/Ks) for soils
without initial ponding described in Table 1, with vertical drainage
(Vachaud) bottom boundary (fw) conditions. Lines represent the SWINGO simplified
model results. Symbols represents Richards equation numerical solution. The
rainfall rate i was selected based on the saturated hydraulic conductivity
(Ks) for each soil to ensure ponding at the surface.
The soil water initial condition in CHEMFLO-2000 was set to hydrostatic
equilibrium with a WT (Eq. 1). The bottom boundary condition was set to a
fixed matric potential h(z=L)=0, to be representative of a WT at depth
L. To simulate rainfall, the top boundary condition is set to a mixed type
boundary with the flux density equal to the specified rainfall rate and the
critical matric potential equalling zero (Nofziger and Wu, 2003). To allow
for the development of distinct tp and tw values
during the simulation, the constant rates of rainfall were chosen based on
the soil texture. This selection was done utilizing a ratio of
i/Ks=6 for the fine soils (clay and silty loam) and
i/Ks=2 for the coarse soils, corresponding to the sandy loam and
fine sandy soils studied by Vachaud and Thony (1971).
Comparison of (a) cumulative infiltration (F) and
(b) wetting front depth (zF) movement results. Lines
represent the SWINGO simplified model and symbols represent Richards equation
numerical solution for soils without initial ponding in Table 1 with vertical
drainage (Vachaud) bottom boundary (fw) conditions.
The comparison of the relative infiltration rates (f/Ks)
calculated by RE (symbols) and the proposed SWINGO (lines) for the case of
vertical drainage end boundary conditions (fw=Ks) is
shown in Fig. 3. The performance of the algorithm is similar to RE for all
soils studied. The median efficiency coefficients Ceff (Nash and
Sutcliffe, 1970) ranged from 0.927–0.9997, with the highest values being for
clay and yielding statistically acceptable models at a 0.01 level of
significance (Ritter and Muñoz-Carpena, 2013) (Table 1). For the same
clay soil with ponded conditions and a WT, Salvucci and Entekhabi (1995)
reported errors of approximately 5 % at time tw, at the point
when the wetting front reaches the WT saturation (zw), and the
infiltration rate switches to the saturated hydraulic conductivity
fw=Ks (f/Ks=1). Smaller differences
(1 % for clay and sandy loam and 3 % for the rest) were found between
both solutions in our tests. These observations indicate that the
simplification (horizontal wetting front, α=0) did not affect the
predictive ability of the infiltration rate. A crucial pattern to notice is
that the estimates of time to ponding acquired across our tested soil types
and normalized rates of rainfall closely matched the outputs of the RE
solution. Our results also indicate that the use of the nonuniform integral
equations (Eqs. 9–12) effectively limit errors in the tp
estimation that sometimes occur when utilizing the Green–Ampt model (Barry et
al., 1996).
Figure 4 displays the cumulative infiltration and the depth of the wetting
front determined using Eqs. (20)–(21) for the vertical drainage boundary
condition for the cases from Table 1. Similar to the infiltration curves,
zF values exhibited a plateau at tw as they reach
column saturation (Fig. 4b), corresponding to the capillary fringe at a depth
of zF=zw=L-hb (Fig. 1), and therefore are
not equal to the depth of the WT (fine sand: L=1.01 m; other soil types:
L=1.50 m).
As the simplified approach is able to produce reliable zF
predictions, it also allows for the depiction of the redistribution of the
soil water content during infiltration. We display the predictions of soil
water (Fig. 5) calculated by the proposed algorithm (dashed lines) compared with the outputs of the RE solution (solid lines) for the
nonponding numerical test examples used previously. The simplified model is
able to identify the midpoint of the wetting front depth at all time points.
Additionally, our simplification of including the horizontal wetting front
(α=0) generates an accurate prediction of soil water at earlier time
points for all soil types, but this prediction decays somewhat at later time
points when approaching column saturation for fine soils. The model does not
degrade at later time points for the sandy soil type when it matches a
horizontal wetting front redistribution. As mentioned previously, because of
the smaller pore space near column saturation, the mass errors generated by
nonzero slopes stay negligible. The infiltration mass balance error at the
end of the simulation (Fig. 4a) ranges from 3 to 8 %. This range of error
values is deemed satisfactory, as these errors are the summation of
approximation errors of both the infiltration and redistribution of soil
moisture generated during the simulation.
Comparison of soil water (θ) redistribution between
Richards equation numerical solution (solid lines) and the SWINGO simplified model
(dashed lines) during infiltration without initial ponding and with vertical
drainage (Vachaud) bottom boundary condition (fw) for soils in Table 1.
Comparison of the simplified and RE results against Vachaud et al. (1974) experimental data set (figure body), and fitting of soil water
characteristics to different equations (inset); vG and Grd represent,
respectively, the van Genuchten and Gardner's soil characteristic curves used
to parametrize the simplified and RE models (see Table 1 for details).
A quick comparison of execution times between CHEMFLO and SWINGO for the
cases in Fig. 5 yielded small reduction of 1–5 s with SWINGO (CPU: 1.6 GHz
Intel Core 2 Duo). These results are machine, computer, and compiler
dependent, where a CHEMFLO finite-difference solution is implemented in Java
computer language (run in Oracle® jre-8u144)
and contains a graphical user interface not intended for optimized
simulations. SWINGO was implemented as a command line application in Fortran
(Intel® Fortran Compiler v17.0.4).
Admittedly, the differences will likely be smaller with optimized code and
new developments of Richards equation implementations (e.g., de Maet et al.,
2014). However, these small differences will likely be compounded in the
context of throughput simulations where the algorithm will be used in some
applications. For example, the model coupled in the companion paper (VFSMOD)
is used in current long-term pesticide regulatory assessments (30-year daily
time steps in the USA or 10-year daily time steps in the EU) (Muñoz-Carpena
et al., 2010, 2015). Considering ∼ 1/3 to ∼ 2/3 of days with
rainfall runoff, the model would be run between ∼ 3000 and
∼ 7000 times for a typical 30 years. assessment. Under this type
of throughput applications condition, even a marginal time improvement can
prove advantageous. In addition to marginal speed benefits, the proposed
algorithm is robust (from the direct integral solution) and has physical
consistency with the original VFSMOD that uses the extension of Green–Ampt for unsteady
rainfall conditions (Chu, 1978; Skaggs and Khaleel, 1982) without the
presence of a shallow water table.
Experimental testing
The physics of the model were tested in a second step using experimental data
from Vachaud et al. (1974) and Chu (1997). The data collected in the
laboratory represents infiltration under ponded conditions in a vertical
column of fine sand soil with a WT at 0.925 m depth. To demonstrate the
generality of the proposed algorithm, the Vachaud et al. (1974) measured soil
hydraulic characteristics were fitted to van Genuchten soil water
characteristic and related unsaturated hydraulic conductivity function based
on Mualem (1976) simplification (vG : vG), and the latter was also fitted to
Gardner function (vG : Grd) (see Appendix A and soil parameters in
Table 1). The goodness of fit of these hydraulic functions (inset of Fig. 6)
shows a small improvement of the K(h) function for Gardner over that of van
Genuchten–Mualem against the experimental data.
The simulated relative infiltration rates obtained with the proposed
algorithm matched the observed data well (Ceff= 0.913–0.942,
RMSE for
f= 5.07 × 10-6–6.20 × 10-6 m s-1),
yielding statistically acceptable models at α= 1 % for vG : Grd
and α= 5 % for vG : vG combinations (Ritter and
Muñoz-Carpena, 2013) (Table 1). The main differences observed between
SWINGO solutions with vG : Grd or vG : Grd soil water functions are
near the time when the wetting front depth approaches the WT, with a small
advance (∼ 0.02 h) introduced by the vG : Grd option. These small
differences are related to the slope of the wetting front being different
than 0, especially close to the intersection with the WT at the end of the
event (Fig. 5). Note also that in this experimental case no observed data
were
available for comparison at the time when the wetting front reached the WT.
Simulation of an unsteady rainfall event on the clay soil in initial
equilibrium with a shallow water table at 150 cm depth, nonponded
conditions, and vertical drainage (Vachaud) bottom boundary conditions
(fw): (a) infiltration (f) and rainfall rates (i, subindices in i1 to
i4 represent the rainfall periods within the hyetograph); (b) cumulative
rainfall (P), infiltration (F), excess runoff (RO), and wetting front depth
(zF) during the event.
Infiltration and excess runoff calculations for an illustrative
unsteady rainfall event on a clay soil with no initial ponding at equilibrium
with a shallow water table at 150 cm depth, smax=0, and end
vertical boundary conditions. The “+” sign in the first column represents
any time right after the beginning of that time step.
Time, t
tp
t0
tw
i
P
f
F
RO
zF
(s)
(s)
(s)
(s)
(m s-1)
(m)
(m s-1)
(m)
(m)
(m)
0
4657.2
2319.0
16 100
2.78 × 10-6
0
2.78 × 10-6
0.0000
0
0
4657.2
4657.2
2319.0
16 100
2.78 × 10-6
0.0129
2.78 × 10-6
0.0129
0
0.017
7500
2.78 × 10-6
0.0208
1.83 × 10-6
0.0192
0.0016
0.253
10 000
2.78 × 10-6
0.0278
1.46 × 10-6
0.0233
0.0045
0.327
10 000+
7.00 × 10-7
0.0278
7.00 × 10-7
0.0233
0.0045
0.327
15 000
7.00 × 10-7
0.0313
7.00 × 10-7
0.0268
0.0045
0.408
15 000+
15 000
13 763.7
18 500
2.78 × 10-6
0.0313
1.21 × 10-6
0.0268
0.0045
0.408
16 500
15 000
13 763.7
18 500
2.78 × 10-6
0.0354
1.08 × 10-6
0.0285
0.0069
0.461
18 000
2.78 × 10-6
0.0396
9.49 × 10-7
0.0300
0.0096
0.535
18 000+
7.00 × 10-7
0.0396
7.00 × 10-7
0.0300
0.0096
0.535
18 500
7.00 × 10-7
0.0399
7.00 × 10-7
0.0304
0.0096
0.569
18 500+
7.00 × 10-7
0.0399
3.40 × 10-7
0.0304
0.0096
0.600
25 000
7.00 × 10-7
0.0445
3.40 × 10-7
0.0326
0.0119
0.600
25 000+
0
0.0445
0
0.0326
0.0119
0.600
Effect of water table depth (L) on cumulative infiltration (F,
represented by isolines) for distinct soils under initial ponding and
different durations of infiltration events (D) for four types of soils and
two end drainage bottom boundary conditions (fw): (a–d) vertical and
(e–h) lateral (So= 0.02, b=1 m, and here Ksh=Ks from Table 1).
In all, these results provide not only a test of the simplified model
against experimental data, but also illustrate its robustness and
flexibility to handle other soil hydraulic functions.
Illustration for unsteady rainfall conditions
The use of SWINGO to simulate realistic unsteady rainfall conditions is
presented for a storm composed of four rainfall periods:
i1= 1 cm h-1 (0 < t≤ 2.8 h),
i2= 0.25 cm h-1 (2.8 < t≤ 4.2 h),
i3= 1 cm h-1 (4.2 < t≤ 5 h), and
i4= 0.25 cm (5 < t≤ 6.9 h) (Table 2 and Fig. 7). The
soil is clay (Table 1) with bottom vertical drainage boundary condition and
smax=0 (i.e., no surface storage). At the beginning of the event
the soil is not ponded and is in equilibrium with the WT at L = 150 cm
below the surface, so max(zF)=zw=L-hb= 0.6 m (Fig. 7). For the initial period, we calculate first the time to
ponding with Eqs. (9)–(12) and (19)
(tp=4657.2 s = 1.29 h), the corresponding t0
(2319 s = 0.64 h) with Eq. (13), and the time to reach the WT
tw (16 100 s = 4.47 h) with Eq. (14). Since the
tw is higher than the rainfall period and tp lower
than the rainfall period, infiltration is equal to the rainfall rate
(f=i1; 0 < t < tp1) before ponding. After
ponding it follows the infiltration capacity curve described by the solution
of Eqs. (16)–(17). At the beginning of the second rain period, since the new
rainfall rate is less than the infiltration rate at the end of the previous
period
(i2=0.25 cm h-1 < fp= 0.52 cm h-1)
and tw is still beyond the period, the infiltration rate equals
the new rainfall rate (f=i2). At the beginning of the third period, the
new rainfall rate is larger than the corresponding potential infiltration
rate at that time
(i3=1 cm h-1 > fp= 0.44 cm h-1)
and ponding starts again immediately such that the new tp=t3
(15 000 s = 4.2 h, beginning of the new rainfall period), and t0
(13 764 s = 3.82 h) and tw (18 500 s = 5.14 h) are
recalculated. Since tw is beyond the period, the infiltration is
maintained at capacity for the duration of this rainfall period. For the last
period, the rainfall rate is lower than the ending infiltration capacity for
the last period (i4= 0.25 cm h-1 < fp= 0.34 cm h-1), and infiltration is initially set to the rainfall
rate. However, since tw is within this period, the soil saturates
when the water front reaches the WT depth (t≥tw), and this
results in saturated vertical drainage flow with unit hydraulic gradient
f=fw=Ks (Eq. 7) until the end of the storm. The
values of the wetting front position (zF) in Table 2 are
calculated from the solution of Eq. (17) during infiltration capacity
(ponding) periods, and the equivalent depths described by Eq. (19) during
nonponding periods. Similarly, cumulative totals are calculated with
Eqs. (3) or (15), and excess rainfall amounts are calculated with the surface
mass balance Eqs. (18) for every time step.
Change in cumulative infiltration (F) as a function of rainfall
rate (i) and water table depth (L) under nonponded initial conditions after a
rainfall duration D= 6 h for the four types of soils in Table 1 and two
end drainage bottom boundary conditions (fw): (a–d) vertical and (e–h)
lateral (So=0.02, b=1 m, and here Ksh=Ks from Table 1). The
isolines describe the change of F with water table depth for the same
rainfall rate (i); hb is the bubbling pressure (capillary fringe) for each
soil type (Table 1).
Evaluation of WT effects on infiltration under conditions of ponding and
nonponding
Figure 8 presents the effect of the WT depth variation (L=0–200 cm) and
event duration (0.5 < D < 6.0 h) on cumulative
infiltration under ponding conditions for the soils in Table 1. The two end-time boundary conditions are compared: fw vertical (a–d) and
fw lateral (e–h) with So= 0.02, b= 1 m, and
Ksh=Ks (Eq. 8 and Table 1). For the conditions tested
it is possible to identify three clearly defined regions (denoted I, II, and
III in Fig. 8) based on the influence of the WT depth on the cumulative
infiltration. Region I (left, shaded in Fig. 8) represents the WT near the
surface, i.e., when it is within the capillary fringe area
L < hb (Fig. 1). The position of the WT in this
region does not affect infiltration since the soil column is already
saturated regardless of L with F=D ⋅ fw. Next,
Region II (clear background in Fig. 8a–d) is the most sensitive to
variations of WT depth, located between L=hb and a limit depth
(L= 125–180 cm) where the variation of F is small (slope less than
0.2 %). This limit depends on the shape of the soil water characteristic
curve for each soil. Finally, Region III represents a region where surface
infiltration can be considered effectively decoupled from the presence of the
WT.
Next, the robustness and physical behavior of the algorithm under nonponded
initial conditions was tested with different rainfall rates
(i= 0.1–20 cm h-1), event durations (D= 1–12 h), and WT depths
(L= 0–400 cm). Figure 9a–d summarizes the results for D= 6 h and
the vertical drainage boundary condition (fw=Ks). Two
main effects are identified. Firstly, as expected F is insensitive to
changes in L for rainfall intensities lower than Ks, when f=i
(no ponding) and F=D ⋅ Ks. Notice that this effect,
although present, is not visible in the clay soil (Fig. 9b) since the
Ks is below the first isoline. Secondly, for rainfall rates above
Ks, the sensitivity to L varies by soil, depending on
hb and the time to ponding values for each rainfall rate
(Eq. 12). As in the ponding case, the soil column is saturated when L≤hb, and there is no sensitivity below this depth. In finer, less
permeable soils (Fig. 9a–b) ponding happens earlier for the same rainfall
rate i, resulting in an increased sensitivity to L with lower rainfall
rates. For the lateral drainage boundary condition, results are similar for
the finer soils (Fig. 9e–f), but much more sensitive to WT depth and
rainfall rate values for more permeable soils (Fig. 9g–h).
Importantly, since excess rainfall runoff is complementary to F
(Eq. 18), these results also quantify the important influence that the
combined effects of WT, soil type, and rainfall intensity can have on surface
runoff flow and transport processes in the VFS.
Summary and conclusions
Limitations in current modeling approaches hamper the evaluation of the
effects of WTs on soil infiltration and runoff. A promising way to overcome these issues is by utilizing simplified
yet realistic specialized algorithms in conjunction with available
hydrological models to evaluate the impact of WTs in the environment.
Previously, Salvucci and Entekhabi (1995) and Chu (1997) recommended the use
of Green–Ampt implicit integral equations to examine infiltration into ponded
soils with WT. We developed and assessed a simplified generic algorithm that
is appropriate for coupling with available hydrological models, in particular
the study of WT effects on VFS runoff pollution control performance. The
proposed SWINGO algorithm is generic – it can utilize any configuration of
soil hydraulic functions – and can be operated under nonponded, ponded, and
realistic variable rainfall conditions to determine runoff (excess rainfall),
infiltration, and soil water redistribution during the event.
SWINGO performed well (Ceff from 0.91 to 0.99) in comparison with the RE
solution and using experimental data on five representative soils. The algorithm
also was able to successfully describe the soil water redistribution during
the simulated event. These useful and reliable predictions indicate that the
proposed approach incorporating a horizontal slope of the wetting front is
suitable for most real-world applications. Through an application of our
proposed SWINGO algorithm, we showed the sensitivity of the infiltration and
excess runoff to the depth of the WT, the length and intensity of the
rainfall event, the soil type, and drainage bottom conditions.
Some of the limitations of the proposed algorithm are the assumptions of a
homogeneous soil profile and horizontal wetting front for fine soils. Future
research is recommended to determine the general validity of the assumption
of a hydrostatic equilibrium and the proposed computation of singular points
during the infiltration episode. Additional experimental testing of the
model should be conducted using data collected under various controlled
and natural conditions (especially during events long enough for the wetting
front to reach the WT). As in real soils a mixture of both end-time lateral
and vertical boundary conditions might be expected, these field studies
could also investigate site and event characteristics for which these
boundary conditions might be dominant or have relative influence.
As SWINGO was accurate, fast and robust when analyzing a variety of
conditions, it is appropriate to couple with currently available
hydrological models to gauge the influence of the presence of WTs on other
landscape processes beyond the simulation of filter strips. The proposed
integral equation has broader relevance as a step forward in improving the
science of hydrologic modeling in general in many other settings, e.g., to
study shallow water table effects on surface runoff, infiltration, flooding,
transport, ecological, and land use processes.
The dynamic coupling with VFS overland flow and sediment and pesticide
transport processes is developed in the companion paper (Lauvernet and
Muñoz-Carpena, 2018). Global sensitivity and uncertainty analysis of the
coupled model is conducted to identify important input factors and their
interactions that will provide better understanding of the fundamental
processes controlling VFS efficiency under WT conditions and guide users to
select effective parameters for practical applications.