A review of stormwater quality and quantity in the urban environment is presented. The review is presented in three parts. The first part reviewed the mathematical methods used in stromwater quantity modelling. This second part reviews the mathematical techiques used in stromwater quality modelling and has been undertaken by examining a number of models that are in current use.

2.1.1. Regression Model

Tasker and Driver (1988) developed simple regression model on the basis of long term urban runoff data and made it applicable for the unmonitored watershed based on some physical (drainage area, impervious percentage, percentage residential or/and industrial) and climatological data (total rainfall, storm duration, mean annual rainfall).1) The model uses the following generalized regression formula for calculating loads:

where, L = pollutant load, X_n = physical, land use or climatic characteristics, β_n ? = regression coefficients, and BCF = Bias Correction Factor

The model parameters are estimated by a generalized-leastsquare regression method that accounts for cross correlation and differences in reliability of sample estimates between sites. The regression models account for 20 to 65 percent of the total variation in observed loads.

2.1.2. Simple Empirical Model

Schueler (1987) introduced an easy empirical equation based model known as Simplified Urban Nutrient Output Model (SUNOM) for urban pollutants load prediction based on five years data collected by United States Environmental Protection Agency (USEPA). The method uses the flow-weighted mean concentration. ²⁾ The generalized equation is as follows:

where L_p = pollutant load, H_r = total rainfall (mm), P_j = percent of rainfall contributes to runoff (equals to 1 for individual storm events), Rv = runoff coefficients estimated as 0.05+0.009^* (impervious percentage), C = flow-weighted pollutant mean concentration (mg/L), A = area (ha)

According to Schueler, the simple method does not consider base flow runoff and associated pollutant loads, and is better used at small watersheds. The model is rarely appered in the literature. Recently, the model was applied by Flint (2004) to estimate water quality an ultra urban area in Maryland, US.³⁾

2.1.3. Sartor and Boyd Model

James Sartor and Gail Boyd first introduced this model in 1972 (Sartor and Boyd, 1972).⁴⁾ This model provides the knowledge of pollutants transport and their quantification. The model shows the dislodging of the particles during a rainfall event is dependent on the street characteristics, rainfall intensity and the particle sizes where the wash off can be described by the following equation:

where, P(t) is the amount of the pollutants washed off in time t, P_o is the initial loading, k = wash-off coefficient, and I = rainfall intensity and t is the time and Q = rainfall volume

Many models such as PSRM-QUAL are based on equation 26 (PSRM-QUAL Users Manual, 1996) and kinematic wave equations.⁵⁾ Once the particle is dislodged the shear forces generated by the runoff cause its movement when the runoff is above the critical velocity (velocity at which drage force and resistance forces are equal). Critical velocity is given by

where V_cr is the critical velocity, C_d is drag coefficient, C_s is static coefficient of friction, g is gravitational acceleration constant, r is the average sediment radius, and SG is sediment specific gravity. United State Environmental Protection Agency (USEPA, 1979) estimated the pollutant load using the following equation.⁶⁾

where, M(t) is the pollutant washoff for time period t (kg), L(t - Δt) is the pollutant accumulation per unit area at the previous time period t - Δt (kg ha^-1), A is the drainage area (ha), and K_w is the watershed washoff coefficient (mm^-1) which is a function of imperviousness of the watershed and the type of simulation, i.e., single event or continuous.

Haiping and Yamada (1996) applied Sartor and Boyd equation with refinement by adding some constants such as i) maximum amount of constituents on impervious areas (k₁) and ii) removal due to wind and vehicles as well as biological and chemical decay (k₂) besides wash-off coefficient.⁷⁾ The amount of pollutant accumulation on impervious surface is given by

where, P_oR is the residual amount of constituents on impervious surface after street sweeping or storm runoff in grams. Residual amount of constituents on impervious areas (P_R) after wash-off in storm is given by;

where, k₃is the wash-off coefficient in mm-1. Q is the total runoff volume (mm). The equation reflects both effects of accumulation in dry weather (Q=0) and wash-off in wet weathers (Q>;0). This can be used as a tool for continuous simulation of urban non point source pollution for a long term prediction.

Furumai et al., (2003) applied Sartor and Boyd model with some modification in urban catchment in Japan. They assumed that the runoff from road and roof are different so that the washoff behavior as follows.⁸⁾ They provided two different runoff coefficients for road and roof.

where, P(t) is the amount of the pollutants washed off in time t, k = wash-off coefficient, and I = rainfall intensity at time t, i = 1 and 2 the roof and road respectively.

The above model (Furumai et al., 2003) was applied by Murakami et al., (2004) to predict the wash-off behavior of particlebound PAHs from road and roof and stated that the model could explain suspended solids and particle-bound PAHs runoff well except during and after heavy rainfall (>;10mm/hr).

Aryal (2003) applied Sartor and Boyd model to predict the pollutants wash-off behavior in highway runoff. As this equation states that the quantity of the constituents available for washoff decreases exponentially with runoff volume during the event, the model could not be applied to the events where the two or more pollutants loading pattern were observed due to the change in rainfall intensity (intermittent rainfall) during wet weather period.⁹⁾ The Sartor and Boyd model found not suitable to the events where two or more SS loading patterns observed. This indicated difficulty in applying the model in those runoff events where the rapid fluctuation of concentration occurred. The following equation establishes the relationship between concentration and the Sartor and Boyd model.

where, C(t) = concentration, quantity/volume F(t) = A^*R(t) = Flow rate (L/sec) A = Area (ha) R(t) = Runoff rate (mm/hr)

This equation (11) shows that the primary difficulty of the Sartor and Boyd equation is that it always produces decreasing concentrations as a function of time regardless of the time distribution of runoff. This is counter-intuitive, since it is expected that high runoff rates during the middle of the storm might produce higher concentration than those proceeding. Aryal (2003) descretized the storm event and applied the model which he finally summed up to calculate the pollutant load.

Egodawatta et al., (2007) also applied the modified version of Sartor and Boyd model by introducing the capacity factor (CF). They reported that a storm event has the capacity to washoff only a fraction of pollutants available and this fraction varies primarily with rainfall intensity, kinetic energy of rainfall and characteristics of the pollutants. They then modified the Sartor and Boyd equation in order to incorporate the wash-off capacity of rainfall by introducing the ‘capacity factor’ CF. According to them, the fraction wash-off can be written as

where, C_F is the value ranging from 0 to 1 depending on the rainfall intensity. Other factors such as road surface condition, characteristics of the available pollutants and slope of the road may also have influence on C_F.

Chen and Adams (2007) also applied the Sartor and Boyd wash-off with refinement by introducing the pollutants accumulation rate based on Osuch-Pajdzinksa and Zawilski (1998) that can accommodate the dry weather period also.^10,11) The rate of pollutant accumulation is:

where M_b is amount of pollutant per unit area on catchment surface, h is the fraction of the impervious area of the catchment, m_d is a constant rate of pollutant deposition (dust fall), m_w is the quantity of street sweeping effectiveness parameter, η is the street sweeping effectiveness parameter, k_b is a constant pollutant removal rate, b is the time elapsed since the last rainfall, β₁  is the conversion of the mass of particulate matter into a parameter of a given type of pollutant and  β₂ describe the conversion of mass of sweeping into a parameter of a given type of pollutant. Integrating the above equation, Mb is:

where, M_o is residual pollutant mass not washed off by the previous runoff event.

In their study they assumed that the rate of pollutant washoff from the catchment surface is proportional to the amount of pollutant build-up on the catchment surface and is directly related to the volume of runoff.

where _r v is the average runoff rate in mm/hr, k_w is the decay or washoff coefficient, in mm^-1. Performing integrated yields

It is the one dimensional conservative advective-diffusion equation, that incorporates the advection and diffusion process is to describe the behaviour of a pollutant in stream;

where, C is the thermal energy or constituent concentration, t the time, x si the distance, u is the advection velocity, Ax the cross-sectional area, D_x is the diffusion coefficient and S (C, x, t) are all sources and sinks.

This equation includes the advection of pollutants by the flowing water, diffusion of pollutants in the stream, constituent reactions, interactions and sources and sinks. Assuming that A_x and D_x are constants and using the flow continuity equation then:

Then

which is the form of the advective-diffusion equation used in model like HEC-5Q and WQRRS.

Shaw et al., (2006) proposed a new stochastic physical model that is primarily focused on the rain flow transportation.¹²⁾ The model was mainly based on Hairsine and Rose (1991) which states that the flow does not exceed the threshold for particles entrainment, mass conservation of suspended particles in the water layer:

where, e is rate of particles enter the shallow flow by raindrop-induced ejection, h is the rate of particle settle-out of the shallow, M_s is the suspended particle mass (g cm^-2), x is the down slope distance, and v is the fluid velocity (cm min^-1)

Particles mass on the surface, M_g (g cm^-2), at a distinct spatial position is given by:

The value e was defined by e = aPM_g

where, a (cm^-1) is an experimentally determined “detachability” constant that accounts for mass loss per drop and P is the precipitation rate (cm min^-1). Similarly particle settling rate is given by (Equation Ommited) where α adjusts bulk concentration to account for variations near the surface, v_set is the particle settling velocity (cm s^-1), and D is the depth (cm).

They also applied water balance at a pint x by using the equation

where, P is the rain intensity per unit width (mL min^-1 cm^-1), q is the flow reate per unit width (mL min^-1 cm^-1) and qo (mL min^-1 cm^-1) is the constant upslope inflow per unit width.

It is another governing one-dimensional equation for pollutant transport on a unit width basis, where solute is injected instantaneously, can be written as

where, C is solute concentration defined as mass of solute per unit volume of water, w is the mass of solute (M) per unit area A of the plane (M/A), w is the mass of pollutant per unit surface area and δ(t)is the instantaneous unit flux of the solute (I/T). The pollutant discharge Q_s is defined as

Qs = CQ

2.4.1. SWMM

SWMM is one of the most successful model produced by United States Environmental Protection Agency (US-EPA). This model is not exclusively designed for urban drainage and single-event or long term (continuous) simulation. The earlier SWMM model used the linear build up formulation. The model provides three options for pollutant build up as follows:

Among the above equation, exponential and Michaelis-Menton functions clearly define asymptotes or upper limits. Upper limits for linear or power function build-up may be imposed if desired. The wash-off equation as follows: using an exponential washoff equation as follows:

where, P(t) is the washoff load rate at time t, K_w is the washoff coefficient, R is the runoff rate (mm/hr), n is the power of runoff rate, and P is the amount of pollutant remaining on the catchment.

As mentioned above earlier, primary difficulty in this equation is always producing decreasing concentrations as a function of time regardless of the time distribution of runoff (Aryal et al., 2003). This problem is overcome in SWMM by making washoff at each time step, P(t), proportional to runoff rate to a power, K_w :

where, P(t) is constituent load washed off at time, t, quantity/sec, P_o (t + Δt) is quantity of constituent available for wash-off at time, t, (e.g., mg), K_w is wash-off coefficient and R(_t) is runoff rate.

It may be seen that if equation is divided by runoff rate to obtain concentration, then concentration is now proportional to ?r^K_w-1. Hence, if the increase in runoff rate is sufficient, concentrations can increase during the middle of a storm even if PSHED is diminished.

From the basic equation (48), the wash-off parameters, washoff coefficient and exponent are determined from a finite difference approximation (Nix, 1994) which produces:¹³⁾

where, P_o (t + Δt) is the amount of pollutant wash-off during simulation time step ((t + Δt), P_o (t)is the amount of pollutant on land surface during a time step (t), Kw is the wash-off or decay coefficient, Δt is the time step, 0.5[R(t)ⁿ + R( t + Δt )ⁿ] is average runoff rate over a time step and n is the power function of runoff rate.

2.4.2. HydroWorks/Infoworks

HydroWorks/InfoWorks calculates the surface pollutant build up for each subcatchment, during a build up (or dry weather) period, before a rainfall event. The basic hypothesis is one of a time-linear accumulation of pollutant on the ground, which depends on the type of activities present on the catchment/subcatchment or in the vicinity. The build-up equation is based on hypothesis that on a clean surface the rate of pollutants accumulation is linear but as the surface mass increases the accumulation rate decays exponentially. The build-up equation is written as:

where, M is mass of the deposit per surface unit (kg/ha), P is build-up factor (kg/ha/day), k_d decay factor (1/day)

The software carries out the following process to determine the build up of pollution for each subcatchment: i. Determine the decay factor ii. Determine the build-up factor iii. Determine the mass of deposit at the end of the build-up period:

where, M_o is the mass of sediment at the end of the build-up period (kg/ha) M_d is the initial mass of deposit in kg/ha (from catchment sediment data (.CSD) file)., k_d is the decay factor (day^-1), N_j is the duration of the dry weather period (days), and Ps is the build-up factor (kgha^-1day^-1).

The surface wash-off model is based, as the runoff module, on the single linear reservoir model. The model consists of sediment erosion and its wash-off. First the amount of sediment eroded from the surface and held in suspension in the storm water is calculated. Then the amount of sediment washed into the drainage system is calculated using a single linear reservoir routing method. The amount of sediment washed into the drainage drainage system is calculated as

where, M_e is mass of sediments dissolved or in suspension per unit active surface (kg/ha), M(_t) is mass of surface deposit pollutants (kg/ha), K_a is erosion/dissolution coefficient (1/s) and is calculated as

where, i(t) is effective rainfall and C₁, C₂ and C₃ are coefficients. Sediment wash-off is given by

where, k is linear reservoir coefficient f(t) is sediment flow Then the sediment flow per subcatchment (kg/s) is calculated (sediment inflow to each node)

where, C is portion of subcatchment, A is subcatchment area.

2.4.3. MOUSE Trap

The MOUSE TRAP model provides several submodules for the simulation of sediment transport and water quality for both urban catchments surfaces and sewer systems. Since pollutants are carried by sediment, the model tries its best to correlate sediment transport process and water quality in sewer systems. Mouse Trap can also model the first flush phenomenon based on temporal and spatial distribution of sediment on the catchment surface and sewer system. Surface Runoff Quality (SRQ) computes the pollutant build-up and transport on catchment surfaces. Two major processes that are involved in SRQ are:

1. Build-up and wash-off of sediment particles on the catchment.

2. Surface transport of pollutants attached to the sediment particles.

2.4.4. MUSIC

MUSIC is one of the most popular stormwater model used in Australia developed by Cooperative Research Centre for Catchment Hydrology (CRCCH) Australia (CRCCH 2005).¹⁴⁾ The model uses simple first order kinetics for the pollutant wash- off from the surface. According to the model the pollutant concentrations in the parcel tend to move by an exponential decay process towards an equilibrium value for that site at that time.

where, C_out is the output concentration, C* is the equilibrium value or background concentration, C_in is the input concentration, k is the exponential rate constant and q is the hydraulic loading (flow rate per surface area) of the treatment measure.

2.4.5. ASTROM

ASTROM model uses the following pollutant build up equation.

where, M_b is the amount of pollutant per unit area on the catchment surface (kg/m²), k_o is constant rate of pollutant deposition (kg/m².h), k_b is constant pollutant removal rate (h^-1), and b is inter-event time.

Integrating above equation yields

where, M_m (= k_o/k_b) represents the maximum amount of pollutant buildup (kg/m²) and M_o is residual amount of pollutant after the previous runoff or street sweeping event (kg/m²).

The pollutant washoff model is defined as

where, l is mass of pollutant washed off per unit area per rainfall event (kg/m²), v_r is runoff event volume (mm) and k_w is pollutant wash-off coefficient.

The model assumes that rainfall event pollutant wash-off load is proportional to, or dependent upon, the accumulated pollutant mass on the catchment surface before the runoff event, and the pollutant wash-off load is a direct function of runoff volume.

Besides, there are several literatures appeared to describe the wash-off behaviour of pollutants during wet weather period. Here, few are described.

Kim et al., (2005) introduced new wash-off model for highway stormwater runoff that incorporates many parameters such as antecedent dry weather periods, rainfall intensity and runoff coefficient.¹⁵⁾ The equation can be initially expressed as

where, C(t) is pollutant concentration; Q(t) is runoff flow rate discharged at time t, α is wash-off rate coefficient, C(t) is pollutant concentration at time t ; V_TRu is total runoff volume (Equation Ommited) which they solved and rearranged in the form

where, M(t) is the pollutant mass emission rate at time t, V_nRu (t) is the normalized cumulative volume, 0≤V_nRu (t)≤1.0

where, δ is an initial concentration related to antecedent dry weather period. The parameters α and γ* are related total runoff. The β* is related to rainfall, runoff coefficient, and storm duration.

This model has two different functions. The first is linear, γ *V_nRu (t)+δ , and the second takes the form of a gamma type function, β *V_nRu (t).Exp[?α .V_nRu (t)]. To use this model it is necessary to predict the total runoff volume, which must be based upon weather forecast or other information.

Kanso et al., (2006) applied simple classical pollutant accumulation followed by the wash-off model to describe the water quality.¹⁶⁾ He described two accumulation behaviours. The first equation calculates the accumulation of pollutants assumed to follow an asymptotic behaviour that depends on two parameters: an accumulation rate D_a (kg/ha/day) and a dry erosion rate D_e (day^-1).

where, M_a(t) (kg) is the available pollutant’s mass at time t and S_i (ha) is the impervious area. The model depends on two parameters: an accumulation coefficient K_a and maximum accumulated mass M₁. It is supposes that the accumulation is proportional to the mass still to be accumulated before reaching the maximum M₁, which is equivalent to the (D_a / D_e).

He described the evolution of the available pollutant mass during the stormwater period by applying the following equation.

(mg/l) is the SS concentration produced by erosion, q(t) is the discharge (m³/s) at the outlet of the watershed at time t, and I(t) is the rainfall intensity (mm/hr).

This paper reviews mathematical methods used in stomwater quality modelling and has been undertaken by examining a number of models that are in current use. The analytical techniques are presented in this paper. The important feature of models is discussed.