This paper presents a three-dimensional (3D) model developed using COMSOL Multiphysics to understand the 3D ex-filtration process of a soak-away rain garden. With a design hyetograph of 3-month average rainfall intensities of Singapore, it is found that the average vertical ex-filtration rate that is obtained by dividing the average vertical ex-filtration (drained through bottom of the soak-away rain garden, averaged over the simulation period = 720 min, and expressed in m3) by the surface area of the soak-away rain garden and the simulation time step is almost constant regardless of increase in saturated hydraulic conductivity (K) of the in-situ soil and the surface area of the soak-away rain garden as a percentage of catchment area. However, as depth to groundwater table which is measured from bottom of the filter media increases, in between 0.5 m and 1 m of depth to groundwater table, the average vertical ex-filtration rate decreases significantly (by around 15 - 20 mm/hr) and the decrease is almost twice, compared with that between 1 m and 1.5 m of depth to groundwater table. Furthermore, this study shows that for a given K of in-situ, K of filter media, and depth to groundwater table, as the surface area of the soak-away rain garden increases, the horizontal flow coefficient which is defined as the ratio between total horizontal ex-filtration (drained through sides of the soak-away rain garden, summed over the simulation period, and expressed in m3) and total vertical ex-filtration (drained through bottom of the soak-away rain garden, summed over the simulation period, and expressed in m3) decreases. Moreover, for a given surface area of the soak-away rain garden, K of in-situ, and depth to groundwater table, the horizontal flow coefficient decreases as K of the filter media increases. However, it is found that for a given surface area of the soak-away rain garden, K of in-situ, and K of filter media, the horizontal flow coefficient increases as depth to groundwater table increases.
To mitigate the adverse impact of urbanization around the world, several best management practices, in other words, green infrastructures, have been used in a way that protect the natural hydrology of the catchment and are more beneficial to the environment [
COMSOL Multiphysics is a powerful interactive environment for modeling and solving partial differential equations in scientific and engineering problems. The software provides a powerful integrated desktop environment with a Model Builder where the users get full overview of the model and access to all functionality. With COMSOL Multiphysics the users can easily extend models for one type of physics into multiphysics models that solve coupled physics phenomena. COMSOL Multiphysics also allows the users to perform various types of studies such as stationary and time-dependent studies. When solving the models, COMSOL Multiphysics uses the proven finite element method. The software runs the finite element analysis together with adaptive meshing and error control using a variety of numerical solvers. A more detailed description of the mathematical and numerical foundation is in the COMSOL Multiphysics Reference Guide [
This section of the paper is divided into three sub-sections. Sub-section 3.1 describes the 3D model development in COMSOL Multiphysics, Sub-section 3.2 briefly summarizes the hyetograph to represent the 3-month average rainfall intensities (ARIs), which was used to simulate the developed model, and Sub-section 3.3 discusses the results.
Modeling of a soak-away rain garden involves flow in variably saturated porous media. Flow in variably saturated porous media is modeled using the Earth Sciences Module (Subsurface Flow Module) of COMSOL Multiphysics. A review of COMSOL’s Earth Sciences Module for simulating flow in variably saturated porous media can be found in [
Using COMSOL Multiphysics, the 3D geometry of the soak-away rain garden of the above mentioned dimensions was formed by first creating a block feature, whose block name is “blk1”, to represent the outer dimensions. The width and the length of this block were set to “15” and the height was set to “5”. Similar to block feature, “blk1”, another block feature, whose block name is “blk2”, was created. The width and the length of this block were set to “5” and the height was set to “1”, but the block position was set to (5, 5, 4) in (X, Y, Z) direction. To represent the in-situ soil, the block feature, “blk2”, was subtracted from the block feature, “blk1”, by creating a Difference feature (“dif1”), a Boolean operation with the 'input' properties set to “blk1” and “blk2”. To represent the filter media, similar to block features “blk1” and “blk2”, another block feature, whose block name is “blk3”, was created. The width and the length of this block were set to “5” and the height was set to “0.6”, but the block position was set to (5, 5, 4) in (X, Y, Z) direction. To represent the soak-away rain garden, the block feature “blk3” was unioned with the Difference feature (“dif1”), by creating a Union feature (“uni1”), a Boolean operation with the 'input' properties set to “blk3” and “diff1”.
Richards’ equation models flow in variably saturated porous media. Many efforts to simplify and improve the modeling of flow in variably saturated media have produced a number of variants of Richards’ equation. In this paper, the form of the Richards’ equation adopted in COMSOL Multiphysics is used [
where the pressure, p, is the dependent variable. In this equation, Cm represents the specific moisture capacity, Se denotes the effective saturation, S is the storage coefficient, κs gives the hydraulic permeability, μ is the fluid dynamic viscosity, kr denotes the relative permeability, ρ is the fluid density, g is acceleration of gravity, D represents the elevation, and Qm is the fluid source (positive) or sink (negative). The fluid velocity across the faces of an infinitesimally small surface is given by Equation (2).
where u is the flux vector. The porous medium consists of pore space, fluids, and solids, but only the liquids move. Equation (2) describes the flux as distributed across a representative surface. To characterize the fluid velocity in the pores, COMSOL Multiphysics also divides u by the volume liquid fraction, θs. This interstitial, pore or average linear velocity is ua = u/θs [
To solve flow in variably saturated porous media, it is necessary that appropriate boundary conditions are specified. From a mathematical standpoint, the application of boundary conditions ensures that the solutions to the problems are self-consistent. In this study, the following boundary conditions are identified as appropriate. As shown in
Water Balance of the soak-away rain garden was carried out using a continuity equation whose components are stormwater runoff, overflow (which is the excess of the ponding space), change of soil moisture within the filter media, change of water storage within the ponding space, vertical ex-filtration, and horizontal ex-filtration. At a given time, horizontal ex-filtration and vertical ex-filtration were computed by integrating the model computed velocity along the four side walls and the bottom surface of the soak-away rain garden, respectively.
The design of soak-away rain gardens involves water quality. Thus, the establishment of a design hyetograph for the design of soak-away rain gardens, specifically, requires data on intensity-duration-frequency (IDF) values for relatively frequent storms such as 3-month ARIs that carry up to 90% of the total load on annual basis. As underscored in the literature, to date, there are few methods available for the establishment of design hyetographs using IDF data [
was considered. Considering an event of 720 min of the 3-month ARIs, a design hyetograph for 3-month ARIs built-up using this method represents a 3-month ARI event both for the 720 min total duration and for any period (i.e., 5 min, 10 min, 15 min, 30 min, 60 min,…, 360 min) within this duration centered on the maximum block [
In this study, as shown in
Furthermore, it is also noted that as depth to groundwater table increases the average vertical ex-filtration decreases. This could be due to the fact that the ex-filtrated flow is inversely proportional to the flow length. Moreover, for a given surface area of the soak-away rain garden (as a % of catchment area) and depth to groundwater table, the average vertical ex-filtration is not influenced significantly as the saturated hydraulic conductivity of the in-situ soil increases. On the other hand, as shown in Figures 6-8, for the considered surface areas of the soak- away rain garden and depths to groundwater table, the average vertical ex-filtration increases as the saturated hydraulic conductivity of the filter media increases. This is due to the fact that as the saturated hydraulic
conductivity of the filter media increases, the ability of the soil fluid to flow through the soil matrix system increases too.
To understand the above discussed behaviors in more detail, a graph of average vertical ex-filtration rate, which is obtained by dividing the average vertical ex-filtration by the surface area of the soak-away rain garden and the simulation time step, versus the surface area of the soak-away rain garden (as a % of catchment area) was plotted. As shown in
As can be observed from the graph, for the considered saturated hydraulic conductivities of the in-situ soil
and for a given depth to groundwater table, the average vertical ex-filtration rate is nearly constant regardless of increase in the surface area of the soak-away rain garden (as a % of catchment area). This indicates that although the average vertical ex-filtrated water through bottom surface of the soak-away rain garden increases as surface area of the soak-away rain garden (as a % of catchment area) increases, the average vertical ex-filtration rate is nearly constant. This could be due to the reason that the incremental/fractional increase on average vertical ex- filtration is approximately equal to the incremental/ fractional increase on surface area of the soak-away rain garden. Furthermore, as shown in
As previously, it is also noted that as depth to groundwater table increases the average vertical ex-filtration rate decreases. However, as the depth to groundwater table increases, in between 0.5 m and 1 m of depth to groundwater table, the average vertical ex-filtration rate decreases significantly (by around 15 - 20 mm/hr) and the decrease is almost twice, compared with that between 1 m and 1.5 m of depth to groundwater table.
To understand the variation of horizontal ex-filtration, as shown in
As can be observed from the graph, for the considered saturated hydraulic conductivities of the in-situ soil, as surface area of the soak-away rain garden (as a % of catchment area) increases, the horizontal flow coefficient decreases. The possible reason for this observation is that as surface area of the soak-away rain garden increases,
it is expected to have more ex-filtrated water. In other words, it is expected that more vertical and horizontal ex- filtration to occur as surface area of the soak-way rain garden increases. However, as surface area of the soak- away rain garden increases, the increase in total vertical ex-filtration is generally higher than the increase in total horizontal ex-filtration. This is owing to the fact that as surface area of the soak-away rain garden increases, the increase in the base area is generally higher than the increase in the area in the side walls. This can be further explained by considering the simulated results of total horizontal ex-filtration and total vertical ex-filtration for a given in-situ hydraulic conductivity and for two different surface areas of the soak-away rain garden (say 4 m * 4 m and 5 m * 5 m). The total horizontal ex-filtration is 1.00E+01 m3 and 1.21E+01 m3 when the surface area of the soak-away rain garden is 4 m * 4 m and 5 m * 5 m, respectively. On the other hand, the total vertical ex- filtration is 6.98E+00 m3 and 1.07E+01 m3 when the surface area of the soak-away rain garden is 4 m * 4 m and 5 m * 5 m, respectively. Thus, the increase in total vertical ex-filtration is 1.53E+00 (1.07E+01/6.98E+00) whe- reas the increase in total horizontal ex-filtration is 1.21E+00 (1.21E+01/1.00E+01). This confirms that as surface area of the soak-away rain garden (as a % of catchment area) increases, the increase in total vertical ex-filtration is generally higher than the increase in total horizontal ex-filtration, and thus has made the horizontal flow coefficient to decrease. Further to this, the declining trend of
Furthermore, for a given surface area of the soak-away rain garden (as a % of catchment area), the horizontal flow coefficient decreases as the saturated hydraulic conductivity of the in-situ soil decreases. The reason for this observation is that as the saturated hydraulic conductivity of the in-situ soil decreases, it is expected that the quantity of ex-filtrated water coming through vertically and horizontally to decrease. However, the amount of decrease for total horizontal ex-filtration is higher compared to the amount of decrease, which is negligible, for total vertical ex-filtration. As shown in
area) and a given saturated hydraulic conductivity of the in-situ soil, as depth to groundwater table increases the horizontal flow coefficient increases. The possible reason for this observation is that as depth to groundwater table goes further down from bottom of the filter media, the flow length increases. However, the flow length is inversely proportional to flow, and thus the total vertical ex-filtration is reduced. Having said this, the total horizontal ex-filtration may not have been influenced by the groundwater table location and has caused the horizontal flow coefficient to increase as the depth to groundwater table increases for a given surface area of the soak-away rain garden and a given saturated hydraulic conductivity of the in-situ soil.
The graphs to represent the saturated hydraulic conductivities of the filter media of 150 mm/hr and 100 mm/hr are placed in
As can be observed from the graphs, for a given surface area of the soak-away rain garden (as a % of catchment area), saturated hydraulic conductivity of the in-situ soil, and depth to groundwater table, the horizontal flow coefficient decreases as the saturated hydraulic conductivity of the filter media increases. The reason for this observation is that as the saturated hydraulic conductivity of the filter media increases, it is expected that the quantity of ex-filtrated water coming through vertically and horizontally to increase. However, the amount of increase for total vertical ex-filtration is higher compared to the amount of increase for total horizontal ex-filtra- tion. The possible reason is that the available area in the side walls for horizontal ex-filtration is lesser than what is available for vertical ex-filtration through the base area of the soak-away rain garden.
This paper presents a 3D model developed using COMSOL Multiphysics to understand the 3D ex-filtration process, which plays major roles in reducing stormwater volume, of a soak-away rain garden, shallow, land- scaped depressions commonly located in parking lots or within small pockets in residential areas. With a design hyetograph of 3-month average rainfall intensities of Singapore, the outcome of this study, among other things, shows the followings:
1) The average vertical ex-filtration rate which is obtained by dividing the average vertical ex-filtration (drained through bottom of the soak-away rain garden, averaged over the simulation period = 720 min, and expressed in m3) by the surface area of the soak-away rain garden and the simulation time step is almost constant regardless of increase in saturated hydraulic conductivity of the in-situ soil and the surface area of the soak-away rain garden as a % of catchment area. However, as depth to groundwater table measured from bottom of the filter media increases, in between 0.5 m and 1 m of depth to groundwater table, the average vertical ex-filtration rate decreases significantly (by 15 - 20 mm/hr) and the decrease is almost twice, compared with that between 1 m and 1.5 m of depth to groundwater table.
2) For a given saturated hydraulic conductivity of the in-situ soil, saturated hydraulic conductivity of the filter media, and depth to groundwater table, as surface area of the soak-away rain garden as a % of catchment area increases, the horizontal flow coefficient which is defined as the ratio between total horizontal ex-filtration (drained through sides of the soak-away rain garden, summed over the simulation period = 720 min, and expressed in m3) and total vertical ex-filtration (drained through bottom of the soak-away rain garden, summed over the simulation period, and expressed in m3) decreases. In other words, as surface area of the soak-away rain garden (as a % of catchment area) increases, the fractional/proportional increase in total vertical ex-filtration is higher than the fractional/proportional increase in total horizontal ex-filtration. Similarly, for a given surface area of the soak- away rain garden, saturated hydraulic conductivity of the in-situ soil, and depth to groundwater table, the horizontal flow coefficient decreases as the saturated hydraulic conductivity of the filter media increases. However, it is found that for a given surface area of the soak-away rain garden, saturated hydraulic conductivity of the in- situ soil, and saturated hydraulic conductivity of the filter media, the horizontal flow coefficient increases as depth to groundwater table increases.
3) Although it is expected that more total vertical and total horizontal ex-filtration to occur as surface area of the soak-away rain garden (as a % of catchment area) increases, there exists a surface area of the soak-away rain garden (as a % of catchment area), beyond which the total horizontal ex-filtration becomes less than the total vertical ex-filtration.
The authors would like to thank PUB, Singapore’s national water agency, for providing the financial support to
conduct this research. The first author, who was a researcher at National University of Singapore, is currently attached to South Eastern University of Sri Lanka.