Figure 1 shows the 3D geometry of a soak-away rain garden that allows stormwater runoff to ex-filtrate into the surrounding soil. For demonstration purpose, a soak-away rain garden with a width of 5 m and a height of 1.0 m

is chosen. The length of the soak-away rain garden is 5 m. The shape of the soak-away rain garden is assumed to be of rectangular/square shape. The height of the filter media, the primary soil layer, is 0.6 m. The height of the ponding space, which is defined above the filter media, is limited to 0.2 m. The width and length of the area of the influence is 15 m. 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 [6] [7] . The Richards’ equation was applied for both the in-situ soil and the filter media of the soak-away rain garden. The adopted equation is presented in Equation (1).

where the pressure, p, is the dependent variable. In this equation, C_m represents the specific moisture capacity, S_e denotes the effective saturation, S is the storage coefficient, κ_s gives the hydraulic permeability, μ is the fluid dynamic viscosity, k_r denotes the relative permeability, ρ is the fluid density, g is acceleration of gravity, D represents the elevation, and Q_m 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 u_a = u/θs [6] [7] .

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 Figure 2, which represents the frontal view of cut-plane (YZ Plane) A-A of Figure 1, the top surface of the rain garden is a rainfall-runoff boundary, a non-steady-state flow condition typical of urban stormwater runoff. The external side boundaries do not allow water to flow in or out of the area of influence, implying that the chosen area is large enough that it does not affect the flow performance around the rain garden. The bottom boundary of the area of influence is specified by a hydraulic head corresponding to an assumed groundwater table level. When water starts to pond, the boundary condition at the top surface of the rain garden becomes a hydraulic boundary. Therefore, there is a need to be able to switch the top surface of the rain garden from flow to hydraulic head. The switching was done using COMSOL Java API, a Java-based interface [6] [7] . The initial condition was set to hydrostatic condition. In other words, above groundwater table, the suction is equal to the distance above groundwater table.

Water Balance of the soak-away rain garden was carried out using a continuity equation whose components are stormwater runoff, overflow (water that is in 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.

To understand the relationship between the dependent variable, overflow volume, and the independent variables, as shown in Figure 4, the graph of overflow volume (total volume of water for the simulation period = 720 min in excess of ponding space)versus the surface area of the soak-away rain garden (as a % of catchment area) was plotted. The overflow volume is expressed as a % of total runoff volume. For this graph, the saturated hydraulic conductivity of the filter media and the depth to groundwater table were set to 100 mm/hr and 0.5 m, respectively. The graph also shows the variation with the saturated hydraulic conductivities of the in-situ soil. The saturated hydraulic conductivity of the in-situ soil varies from 10 mm/hr to 50 mm/hr. As can be observed from the graph, there exists a linear relationship between the dependent variable and the independent variables. This same behaviour was observed with the other graphs, which account for the variation in depth to groundwater table and the saturated hydraulic conductivity of the filter media, as well. Therefore, it was decided to fit a multiple linear regression. In the subsequent paragraphs, the statistical measures of the regression are discussed.

As shown in Table 1, before fitting a multiple regression, the simulated results for all the considered ranges of potential hydrologic conditions or independent variables (e.g., size of the soak-away rain garden, saturated hydraulic conductivity of the in-situ soil, saturated hydraulic conductivity of the filter media) were analyzed to develop a correlation matrix to detect if the problem of multicollinearity exists. The problem of multicollinearity, which can have impact on the quality and stability of the fitted regression model, occurs when some of the independent variables are highly correlated.

In Table 1, dependent variable, “Y”, is the overflow volume which is expressed as a % of total runoff volume. X₁, X₂, X₃, and X₄, which are the independent variables of the regressed equation, are saturated hydraulic conductivity of the filter media, saturated hydraulic conductivity of the in-situ soil, depth to groundwater table measured from bottom of the filter media, and size of the soak-away rain garden (as a % of catchment area), respectively. The correlation matrix shows that there is essentially no correlation between the independent variables and dependent variable, and between the independent variables themselves. Thus, as the problem of multicollinearity does not exist, the regression relationship was further analyzed. As placed in Table 2, the sample multiple correlation is 0.996. It has a very strong positive correlation. Table 2 also shows the coefficient of determination, which in this case is 0.992, indicating that the proportion of the variation in Y that is due to independent variables in the sample is 0.992. In other words, there is very little unexplained (residual) variation whose mean square error is only 1.182, as shown in Table 3.

On the other hand, the test of significance is placed in Table 3. If the independent variables explain very little of the Y, dependent variable, the F-value should have been 1 or less. However, the fitted regression is very significant as the F-value is 6881.361. In the fitted regression, the explained variation due to regression and the independent variables is 6881.361 times greater than the unexplained variation. Furthermore, if the selected independent variables and Y are unrelated, the probability of getting the sample evidence is more extreme and is 6.3702E-234. In other words, it is almost impossible to get this kind of data as a result of chance and thus the null hypothesis is rejected. Thus, it is concluded that there is a relationship between these variables in the population.

The “weight” given to each independent variable in the fitted regression is presented in Table 4. In a multiple linear regression, the value of the coefficient for each independent variable gives the size of the effect that variable is having on the dependent variable, and the sign on the coefficient (positive or negative) gives the direction of the effect. In the fitted regression, except the independent variable, X₃, depth to groundwater table measured from bottom of the filter media, the considered independent variables have a negative effect on dependent variable, overflow volume which is expressed as a % of total runoff volume. When these values of coefficients are placed in the regression equation, the equation for predicting the overflow volume, which is expressed as a % of total runoff volume, from the considered independent variables, is Y = 74.441 − 0.122X₁ − 0.105X₂ + 14.672X₃ − 5.224X₄.

The t-test results to test the regression for significance are also presented in Table 4. As can be observed from Table 4, the t-values for independent variables are very significant. The probabilities of getting of these magnitudes if the null hypothesis for the test is true are very small as given by p-values. Thus, it is almost impossible to get this kind of data as a result of chance and thus the null hypothesis is rejected. Therefore, it can be concluded that fitted regression between these variables is significant.

To understand the relationship between the dependent variable, horizontal flow coefficient, and the independent variables, as shown in Figure 5, the graph of horizontal flow coefficient versus the surface area of the soak- away rain garden (as a % of catchment area) was plotted. Horizontal flow coefficient 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). For this graph, the saturated hydraulic conductivity of the filter media was set to 100 mm/hr and the depth to groundwater table measured from bottom of the filter media was set to 0.5 m. The graph also shows the variation with the in-situ hydraulic conductivities. As can be observed from the graph, there exists a non-linear (power: Y = aX^b) relationship between the dependent variable and the independent variables. This same behaviour was observed with the other

graphs, which account for the variation in depth to groundwater table and the saturated hydraulic conductivity of the filter media, as well. Thus, it was required to transform the data using a log-log transform. Having transformed the data, a multiple linear regression to find the best-fit line for the transformed data was obtained. In the subsequent paragraphs, the statistical measures of the regression are discussed.

Table 5 shows the correlation matrix. As shown in Table 5, the dependent variable, “Y”, is the log of horizontal flow coefficient. X₁, X₂, X₃, and X₄, which are the independent variables of the regressed equation, are the logs of saturated hydraulic conductivity of the filter media, saturated hydraulic conductivity of the in-situ soil, depth to groundwater table measured from bottom of the filter media, and size of the soak-away rain garden (as % of catchment area), respectively. The correlation matrix shows that there is essentially no correlation between the independent variables and dependent variable, and between the independent variables themselves. Thus, as the problem of multicollinearity does not exist, the regression relationship was further analyzed.

As shown in Table 6, the coefficient of determination is 0.986 and thus there is a very little unexplained variation. Furthermore, as shown in Table 7, the fitted regression is very significant and the explained variation due to regression and the independent variables is 5498.676 times greater than the unexplained variation. Furthermore, if the selected independent variables and Y are unrelated, the probability of getting the sample evidence is more extreme and is 2.2218E-286. Thus, it was concluded that there is a relationship between these variables in the population.

The “weight” given to each independent variable in the fitted regression is presented in Table 8. When these values are placed in the regression equation, the equation for predicting log of horizontal flow coefficient, from logs of the considered dependent variables, is Y = 1.318 − 0.236X₁ + 0.160X₂ + 1.569X₃ − 0.552X₄. The t-test results to test the regression for significance are also presented in Table 8. As can be observed from Table 8, the t-values for independent variables are very significant. The probabilities of getting of these magnitudes if the null hypothesis for the test is true are very small as given by p-values. Thus, it is almost impossible to get this

kind of data as a result of chance and thus the null hypothesis is rejected. Therefore, it can be concluded that fitted regression between these variables is significant.

To understand the relationship between the dependent variable, average vertical ex-filtration rate, and the independent variables, as shown in Figure 6, the graph of average vertical ex-filtration rate was plotted. The average vertical ex-filtration rate is obtained by dividing the average vertical ex-filtration (drained through bottom of the soak-away rain garden, averaged over the simulation period, and expressed in m³) by the surface area of the soak-away rain garden and the simulation time step. For this graph, the saturated hydraulic conductivity of the filter media was set to 100 mm/hr. The graph also shows the variation with the in-situ hydraulic conductivities. The saturated hydraulic conductivity of the in-situ soil varies from 10 mm/hr to 50 mm/hr.

As can be observed from the graph, there exists a linear relationship between the dependent variable and the independent variables. This same behaviour was observed with the other graphs, which account for the variation in saturated hydraulic conductivity of the filter media. Therefore, it was decided to fit a multiple linear regression. In the subsequent paragraphs, the statistical measures of the regression are discussed. As shown in Table 9, by performing a normal multiple regression using all the independent variables, it was found that X₂, saturated hydraulic conductivity of the in-situ soil, and X₄, surface area of the soak-away rain garden (as % of catchment area) have high p-values (p-values > significance level, α = 0.01) even though the “Significance F” is very low. Thus, forward stepwise regression was used to develop the best model that contains some of the independent variables.

With forward stepwise regression, as shown in Table 10, the correlation matrix shows that there is essentially no correlation between the independent variables and dependent variable, and between the independent variables themselves. Thus, as the problem of multicollinearity does not exist, the regression relationship was further analyzed.

As shown in Table 11, the coefficient of determination is 0.911 and thus there is a very little unexplained variation. Furthermore, as shown in Table 12, the fitted regression is very significant and the explained variation due to regression and the independent variables is 1591.399 times greater than the unexplained. If the selected independent variables and Y are unrelated, the probability of getting the sample evidence is more extreme and is 2.062E-164. Thus, it was concluded that there is a relationship between these variables in the population.

The “weight” given to each independent variable in the fitted regression is presented in Table 13. When these values are placed in the regression equation, the equation for predicting theaverage vertical ex-filtration rate from the considered independent variables, is Y = 26.682 + 0.077X₁ − 23.483X₃.

The t-test results to test the regression for significance are also presented in Table 13. As can be observed from Table 13, the t-values for independent variables are very significant. The probabilities of getting of these magnitudes if the null hypothesis for the test is true are very small as given by p-values. Thus, it is almost impossible to get this kind of data as a result of chance and thus the null hypothesis is rejected. Therefore, it can be concluded that fitted regression between these variables is significant.