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BC TRCR – September 2021
AN INVERSE SOIL-PLANT-ATMOSPHERE MODEL TO ESTIMATE VEGETATION AND
HYDRAULIC PROPERTIES OF MATERIALS FROM FIELD MEASUREMENTS TO
IMPROVE PERFORMANCE PREDICTIONS AND CONFIDENCE IN MINE RECLAMATION
R.E. Shurniak, P.Eng.
Okane Consultants
112 – 112 Research Drive
Saskatoon, SK V9R 6X1
ABSTRACT
Cover system and landform design are crucial components of mine reclamation, of which the hydraulic
performance is largely controlled by vegetation and material properties. Field monitoring (especially
volumetric water content profiles) of a reclaimed area provides valuable information on a cover system’s
and/or landform’s performance and indirectly measure a site’s vegetation and material properties. One of
the most challenging and time-consuming numerical modelling problems is calibrating a model for near-
surface (i.e., vadose zone) water movement, which accounts for atmospheric, vegetation, and material
interactions (referred to as a Soil-Plant-Atmosphere or SPA model), to such field measurements.
Previously, establishing inputs for vegetation and hydraulic properties required the model user to estimate
initial inputs and then subjectively adapt the inputs while completing a multitude of iterations until a
reasonable fit to field conditions was realized. To improve this process, the author has developed a
simplified SPA model using the GoldSim software to quickly and objectively develop estimates of
vegetation and hydraulic properties. The vegetation and hydraulic properties can then be validated within
more rigorous and recognized SPA modelling software. The process can also be tailored to identify
potential equifinal solutions, which can usually be eliminated through multiple factor calibration. The
results of the model are calibrated vegetation and hydraulic properties that can be used to confidently predict
mine reclamation performance. The paper provides an example where this modelling process has been
successfully applied for a reclaimed site in the Northern Hemisphere.
Key Words: calibration, modelling, cover systems, reclamation, landform, vegetation
INTRODUCTION
The development and calibration of a Soil-Plant-Atmosphere (SPA) numerical model capable of simulating
measured field responses (referred to hereinafter as a field response model) is a vital component of the
cover system and landform design processes for mine reclamation. Once calibrated, inputs from a field
response model are powerful tools that can be used in performance models to evaluate long-term cover
performance for an extensive variety of soil, plant, and climate scenarios to aid in finalizing a cover design.
However, determining a set of calibrated field response model inputs is a complex and challenging process,
requiring many iterations before model results and field measurements converge. In the past, this iterative
process had to be completed manually, which introduced a lot of subjectivity into the model inputs and a
higher potential for equifinality (Beven and Binley, 1992; Beven and Freer, 2001; Beven, 2006), a condition
where different model inputs may lead to equally acceptable representations of field responses. To simplify
this process and improve objectivity, an inverse SPA model is presented in this paper.
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Inverse models use the effects of a problem to calculate the causes. For this paper, stored water volumes
and volumetric water content (VWC) measurements (i.e., the effects) are used to calculate soil hydraulic
properties and rudimentary vegetation properties (i.e., the causes). The history of this type of inverse model
(referred to as an inverse SPA model in this paper) has been well documented in previous papers by Ines
and Droogers (2002) and, more recently, Li et. al. (2017). In most past cases, the inverse SPA model is
created by coupling an existing SPA model with a program that finds the best solution to an optimization
problem (most commonly PEST (Doherty et al., 1994)). Although this method shows great promise, it is
also highly complex, as it requires the model user to know to a high level of sophistication both the SPA
and optimization modelling products as well as how to program their linkage. Hence, the author has
developed a simplified SPA model using the GoldSim v12.1 software (GoldSim, 2018), which, by using
GoldSim’s built-in optimization module, can quickly and objectively develop initial estimates of vegetation
and hydraulic properties. These estimated properties can then be validated within a more rigorous and
recognized SPA modelling software.
MATERIALS AND METHODS
The inverse SPA model developed within the GoldSim software for this paper uses the finite-difference
method, in which both the vertical layering and time intervals are discretized so that the Richards equation
(Richards, 1931), a nonlinear partial differential equation (PDE), is simplified into a system of algebraic
equations. The Richards equation for vertical water flow within a soil profile is:
()/=/ (()(/+ 1)) [1]
where θ(h) is volumetric water content (L3L-3) as a function of pressure head, h (L); t is time (T); z is the
vertical spatial coordinate (L); and k(h) is the hydraulic conductivity (LT-1) as a function of h.
To solve Equation [1], hydraulic properties (namely, a water retention curve, θ(h), and hydraulic
conductivity function, k(h)) need to be defined for each layer of material. For the modelling presented in
this paper, the van Genuchten (1980) equations were used to define both hydraulic functions:
()= +
[||] < 0
0
[2]
()=11
[3]
where θr and θs refer to the residual and saturated volumetric water contents (L3L-3), respectively; Se is the
effective saturation, defined as (θ – θr)/(θs – θr); α (L-1), n, and m are van Genuchten’s equation parameters,
where m = 1 – 1/n; ks is the saturated hydraulic conductivity (L T-1); and l is the pore-connectivity parameter,
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which Mualem (1976) estimated to be 0.5, on average, for many soils, but was allowed to range between 3
and -3 for the work completed for this paper.
To simulate the surface boundary condition, precipitation (if any) must be applied, runoff (if any) removed
and actual evaporation (AE) and (in the case of vegetation growth) transpiration (AT) estimated. As the
model described in this paper does not account for ground freezing, it is only used to simulate fully thawed
conditions; hence, precipitation is a direct input of daily rainfall (or irrigation) measurements. Additionally,
to simplify the model, all precipitation is assumed to be directly added to the stored water volume of the
surface material layer of the model; however, if this results in more water than the surface layer can hold
then the excess is reported as runoff. Hence, the model does not account for Horton (1933) overland flow
(i.e., when rainfall exceeds infiltration capacity of the surface material), only saturated overland flow.
These assumptions could be changed quite easily in the GoldSim software if warranted by the situation
being simulated.
Wilson et al. (1997) presented laboratory results that indicated a unique relationship between the ratio of
actual evaporation to potential evaporation (AE/PE) and total suction (matric suction plus osmotic suction)
that appears to be independent of soil texture, drying time, and water content. Assuming air, water, and
soil are at approximately the same temperature, they also developed an equation that passed through the
experimental data with a reasonable fit. Based on this work, Fredlund et al. (2011) developed the following
equation for AE/PE which was used in the inverse SPA model:
=exp (()
()) [4]
where ψm is matric suction (kPa); ψo is osmotic suction (kPa); g is acceleration due to gravity (m.s-2); ωv is
the molecular weight of water (0.018 kg.mol-1); ϕ is the air relative humidity; γw is the unit weight of water
(9.807 kN.m-3); R is the universal gas constant (8.314 J.mol-1.K-1); T is the absolute temperature of the soil
surface (K), but assumed to be the same as the absolute air temperature for this paper, and; ζ is a
dimensionless empirical factor suggested to be 0.7. It is assumed that the pressure head calculated for the
surface layer of the model is the matric suction in Equation 4, and the osmotic suction is estimated by the
inverse SPA model during the parameter optimization process; therefore, the osmotic suction is treated as
a calibration factor by the model.
Using Equation 4, AE can be estimated by multiplying AE/PE by PE, which is either measured or estimated.
To account for vegetation water use, a simplified method for estimating AT was added to provide an
estimate for actual evapotranspiration (AET) which was programmed into the inverse SPA model:
=(
(100% )+(()1
+()1
)) [5]
where PE is potential evaporation (LT-1); Cover is assumed to represent the percentage of the ground
surface covered with a vegetation canopy (%); PWL(h) is the plant water limiting function value at the
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current pressure head (h) of the surface layer of the model; h1 and h2 are the pressure heads of the surface
layer and the layer below the surface layer (L), respectively; R1 and R2 are the average depths of the surface
layer and the layer below the surface layer (L), respectively; and RT is the total simulated rooting depth,
which is assumed to be the total thickness of the surface layer and the layer below the surface layer
combined (L). The plant water limiting function is assumed to have a value of one at or above field capacity
and zero at or below its permanent wilting point; a linear relationship between these two points defines the
remainder of the function. For the modelling presented in this paper, field capacity is assumed to occur at
a matric suction of 10 kPa and permanent wilting point at a suction of 1,500 kPa. Note that the rooting
depth defined with Equation 5 assumes rooting is only established in the surface two layers of the model
and that a triangular root distribution establishes. However, as stated with previous assumptions, The
GoldSim model can be adapted to account for additional rooting depth and/or varied root distributions if
warranted by the situation being simulated.
To optimize the parameters being evaluated by the inverse SPA model, the optimization module within
GoldSim runs a multitude of iterations to minimize the following residual sum of squares (RSS):
= ()
[6]
where Mt is the modelled stored water volume within the simulated material profile at time t, and; Ot is the
observed stored water volume at time t.
The coefficient of determination (R2) is used to provide a measure of how well the observed stored water
volumes are replicated by the model:
= 1
[7]
where TSS is the total sum of squares:
= ()
[8]
and Ō is the average observed stored water volume.
SIMULATION OF A LABORATORY EXPERIMENT
Huang et al. (2012) constructed five soil columns in a laboratory setting, three layered soil columns and
two of homogenous soil columns of sand and silt, to evaluate, as the title of their paper implied,
“Evaporation and Water Redistribution in Layered Unsaturated Soil Profiles”. Each column was carefully
monitored for changes in water storage, lower boundary fluxes, and potential and actual evaporation with
time. The hydraulic properties of the sand and silt were measured but also verified through simulations of
the homogeneous sand and silt columns. The silt was also used as an example to analyze the sensitivity of
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the simulated cumulative lower boundary flux to the hydraulic conductivity function. Hence, the silt
column was simulated for this paper to provide initial validation of the inverse SPA model.
Description of Silt Column Test
As described in more detail in Huang et al. (2012), the silt column was 600 mm in height with a 127 mm
inner diameter. The column base was constructed with drainage grooves to permit drainage from the entire
soil profile at a lower outlet, which could also be used to create a constant head water table condition as
required.
The silt column evaporation test was conducted in a chamber maintained at 36 ± 1.0°C to accelerate
evaporative fluxes. The average daily relative humidity was 11.1 ± 3.8%. PE for the silt column test was
measured at 5.6 ± 0.4 mm d-1. The testing was completed in two, 31-day periods: during the first period, a
water reservoir was connected to the outlet port to create a water table with an elevation of 5 mm above the
base of the soil column. The water reservoir was then removed, and the outlet port sealed to establish a
zero-flux lower boundary condition for the second, 31-day period.
Water content profiles were measured at elapsed times of 1, 3, 5, 7, 10, 14, 21, 31, 32, 34, 38, 45, and 61
days. AE was obtained by independently measuring the inflow volume and the mass of the column using
a triple balance beam.
Description of Silt Column Models
The inverse SPA model was programmed into GoldSim as described in the ‘Materials and Methods” section
of this paper. The model was programmed to divide the column into four elements that were each a
thickness of 150 mm. Each layer was assumed to have the same hydraulic properties.
When running the optimization module in GoldSim, the parameters were allowed to vary as presented in
Table 1. The optimization attempted to minimize the RSS when comparing the measured and simulated
total stored water volume within the silt column with time.
The computer program GeoStudio 2021.3 (GEOSLOPE, 2021) was used to validate the parameters
estimated by the inverse SPA model. The GeoStudio model was set up as a one-dimensional model
consisting of four regions of 150 mm thickness and each region was then subdivided into 6 elements; hence,
24 elements total.
Both models were set to start their simulations at the end of the first day of the experiment when the first
water content profile was measured, and end 60 days later (i.e., 61 days after the start of the experiment)
when the final water content profile was measured. The lower boundary was set as a pressure head equal
to 5 mm for the first 30 days of the simulation, and then changed to a zero-flux boundary for the last 30
days. The air temperature, relative humidity, precipitation, and PE were all estimated to remain constant
throughout the 60-day simulations at their average measured daily values of 36°C, 11.1%, 0 mm d-1, and
5.6 mm d-1, respectively.
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Silt Column Model Results
After completing thousands of iterations, the inverse SPA model provided the set of hydraulic parameters
listed in Table 1. As shown in Figure 1, these parameters define a water retention curve (a) and hydraulic
conductivity function (b) similar to those estimated by Huang et al. (2012), but with two distinct differences:
the residual water content and the slope of the hydraulic conductivity function. As explained in the
following paragraphs, it is hypothesized by the author that these differences are mainly due to the column
being only 600 mm in height, having artificial lower boundary conditions, and a relatively short duration
for the total experiment.
Table 1. Ranges and optimized parameters estimated to simulate the silt column.
Parameter
Optimization Range
Inverse SPA Model
Huang et al. (2012)
θs (m
3
m
-3
)
0.4 to 0.5
0.415
0.408
θr (m3 m-3)
0.0 to 0.3
0.2700
0.0095
α (m
-1
)
9.8E-2 to 9.8
0.392
0.270
n
1.1 to 15
4.800
3.082
l
-3 to 3
-1.63
0.5
ks (m
.
s
-1
)
1E-10 to 1E-7
5.00E-9
5.65E-9
Figure 1. Measured and estimated (a) water retention curves and (b) hydraulic conductivity functions.
Measured water retention curve from Bruch (1993).
Firstly, because the column started the experiment tension saturated with an artificial water table established
for the first 31 days, the volumetric water content profile within the column was above the residual water
content estimated by the inverse SPA model for most of the experiment. As a result, the calibration was
weighted more heavily to correctly estimate the hydraulic properties at low suctions, and the inverse SPA
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model ensured a high degree of saturation was maintained by estimating a high residual volumetric water
content. Had the experiment gone on for longer, the inverse SPA model would have put more weight on
developing a calibration at higher suctions. This indicates a potential weakness when using the inverse
SPA model, and the need to calibrate to a system with well-defined wetting and drying events.
Secondly, Huang et al. (2012) reported they were unable to accurately simulate the silt column for the first
31-day period when just assuming a constant head lower boundary. It was suggested (based on work by
Willis (1960)) “…that this could be due to entrapped air, which restricts upward water movement from the
lower water table boundary, and differences in the degree of packing between the soil in the column and
the soil sample used in the hydraulic conductivity tests.” As a result, they resorted to simulating the lower
boundary condition by applying the measured lower boundary fluxes. However, as shown in the results
provided in Figure 2 and Table 2, the hydraulic inputs developed by the inverse SPA model were able to
reasonably simulate the water balance for the silt column for the duration of the experiment (R2 = 0.88),
even for the first period when a constant head lower boundary was used instead of applying the measured
lower boundary fluxes as was done by Huang et al. (2012). Hence, it is hypothesized that the inverse SPA
model developed hydraulic properties for the silt that also accounted for any unintended and unaccounted
for issues with the experiment set-up. It is also assumed that changing the lower boundary on day 31 also
changed some of the unintended and unaccounted for issues, which is why the simulated water balance
diverged from the measurements. This indicates that it is best to use the inverse SPA model tool to simulate
a system where the edge boundaries are minimally impacting performance.
Figure 2. Measured and simulated water balance components with time for the silt column
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Table 2. Measured and simulated water balance components and experimental errors for the silt column
Lower Boundary
Inflow (mm)
Actual Evaporation
(mm)
Change in Soil
Water Storage (mm)
Absolute Error
(mm)
Measured
57.5
139.0
-72.0
9.5
Simulated
58.6
144.5
-87.8
1.9
Even given these issues, the inverse SPA model showed great promise as it provided a reasonable estimate
of the column’s water balance and hydraulic properties – especially the silt’s ks using only the total stored
water volume, climate, and lower boundary conditions as inputs. Hence, a field monitoring site was chosen
to provide a second example of the inverse SPA model’s utility.
SIMULATION OF A FIELD SITE
Description of the Field Site
The chosen field site is located in the northern hemisphere on a northeast-facing slope, 1,600 masl, where
a cover system consisting of 25 cm of regolith material overlying waste rock is being monitored. A profile
of eight thermal conductivity sensors and eight time-domain reflectometry sensors were installed to monitor
the matric suction and volumetric water contents at depths of 5 cm, 10 cm, 20 cm, 30 cm, 40 cm, 50 cm,
75 cm, and 100 cm. A meteorological station was installed on the site to monitor rainfall, air temperature,
relative humidity, wind speed and direction, net radiation, and snow depth. Grasses were actively growing
on the site for the period selected for simulation (July 1st to August 31st). Leaf area index (LAI) and rooting
depth were measured at site during mid-September.
Description of the Site Model
The inverse SPA model used the same four-layer framework as was used for the silt column model. The
top two layers were simulated using a thickness of 25 cm each, with the underlying (third) and base (fourth)
layers simulated as 50 cm and 400 cm thick, respectively. Hence, the top three layers were used to calibrate
to the measured total stored water volume in the top 100 cm of the field site (estimated from the volumetric
water content measurements). The top two layers were simulated with a combined thickness of 50 cm to
match the measured thickness for the bulk of the root biomass. The base layer was used to provide a
sufficient layer of waste rock between the monitored profile depth and the lower boundary condition so that
the lower boundary would have minimal influence on the model performance. The regolith material was
found to have a similar particle size distribution to the waste rock, so all the layers were assumed to have
the same hydraulic properties.
Table 3 provides the ranges for the hydraulic parameters and vegetation “cover” parameter that were used
when running the optimization module in GoldSim. The optimization attempted to minimize the RSS when
comparing the measured and simulated total stored water volume within the top 100 cm of the field site
with time.
BC TRCR – September 2021
As with the silt column simulations, the GeoStudio program was used to validate the parameters estimated
by the inverse SPA model. The GeoStudio model was set up as a one-dimensional model consisting of four
regions matching the thicknesses of the four layers defined in the inverse SPA model. A global element
size of 5 cm was used to subdivide the 500 cm profile into 100 elements. To match the inverse SPA model’s
estimated vegetation inputs, the rooting depth was set at 50 cm, with a triangular root distribution. the
vegetation “cover” estimate from the inverse SPA model was used to estimate LAI using the following
formula based on the work of Tratch (1996):
=
.
. [9]
Both models were set to start their simulations on June 1st and end on August 31st (i.e., 92 days). The air
temperature, relative humidity, and precipitation measurements taken at the meteorological station during
the chosen simulation period were used directly in the models. PE was estimated from the measured climate
data using the Penman-Monteith method (as described by Allen et al., 1998).
The lower boundary for both models was set as a unit hydraulic gradient. A unit hydraulic gradient
boundary condition assumes that at the lower boundary the pressure head (and, as a result, water content
and hydraulic conductivity) are constant with depth. When this is the case, the total head equals the
gravitational head causing a unit hydraulic gradient. In other words, a unit hydraulic gradient represents a
location in the modelled profile where water movement is controlled mainly by gravity.
Site Model Results
The inverse SPA model converged on the input parameters provided in Table 3. Using these parameters in
the GeoStudio model resulted in the water balance results provided in Figure 3, which shows that the input
parameters created an excellent fit to the measured change in storage for the monitored profile (R2 = 0.83).
As shown in Figure 4, the water retention curve defined by the input parameters also fits well to the field
water retention data (obtained by plotting the matric suction and volumetric water content data measured
at the same depths).
Table 3. Ranges and optimized parameters estimated to simulate the field site.
Parameter
Optimization Range
Inverse SPA Model Optimized Values
θs (m3 m-3)
0.2 to 0.3
0.231
θr (m
3
m
-3
)
0.0 to 0.15
0.056
α (m-1)
9.8E-2 to 9.8
2.145
n
1.1 to 5
1.15
l
-3 to 3
3
ks (m
.
s
-1
)
1E-8 to 1E-4
6.52E-5
Cover (%)
0% to 100%
90%
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Figure 3. Measured and simulated water balance components with time for the field site.
Figure 4. Field measured water retention data and simulated water retention curve used for the site model.
-100
0
100
200
300
400
1-Jun
1-Jul
1-Aug
1-Sep
Water Balance Components (mm)
Rainfall
Simulated Actual Evapotranspiration
Simulated Net Percolation
Simulated Runoff
Simulated Change in Storage
Measured Change in Storage
330 mm
249 mm
139 mm
0 mm
-57 mm
-59 mm
R
2
= 0.83
BC TRCR – September 2021
As presented in Table 3, the inverse SPA model estimated 90% of the ground surface was covered with
vegetation, which is calculated to represent a LAI of 2.51 using Equation 9. This falls within the LAI range
measured for the site in mid-September of 2.16 to 2.57. As LAI is likely declining in mid-September at this
location, it is assumed that these measurements are slightly lower than they would have been during the
June to August simulation period, which provides more credence to the accuracy of the inverse SPA
model’s LAI estimate.
CONCLUSION
With only climate data and data from a profile of volumetric water content sensors, the inverse SPA model
can provide reasonable initial estimates of hydraulic and rudimentary vegetation properties for a cover
system’s materials and/or materials being reclaimed with little modelling time or effort. The need for a
complimentary profile of suction sensors is reduced as the model can achieve a calibration without this data
(however, if available, the suction data can be used for additional model validation). The inverse SPA
model can be readily adapted to provide an initial estimate for almost any location, with the estimated
parameters easily transferred and tested in a recognized SPA modelling software. Additionally, the ranges
for parameters being tested and output conditions can be adapted as additional information becomes
available, or to test the uniqueness of the initial solution. As with any model, and as shown by the Silt
Column Model results, care must be taken to not give too much credence to the parameters estimated by
the inverse SPA model until they have been thoroughly vetted and validated. However, once validated,
the calibrated vegetation and hydraulic properties can then be used to evaluate current and future
performance with high confidence.
Having the inverse SPA model programmed within the GoldSim software makes it easily adaptable to
tackle more complex problems such as multi-layered cover systems, evaluating spatial variations, and
evaluating how a system evolves with time. Future planned additions to the model include:
• Accounting for multi-modal water retention curves and hydraulic conductivity functions; and
• Coupling the GoldSim software directly to an existing SPA model.
The inverse SPA model is a valuable tool for gaining quick understanding of a system’s hydraulic
performance. Calibrations that used to take weeks or months to achieve can now be realized in hours.
AKNOWLEDGEMENTS
Thank you to Dr. Lee Barbour for providing the experimental data from their column experiments. Your
ongoing support is much appreciated and gratefully acknowledged.
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