In its present form, the model uses a simplified version of the continuity equation to route stream flow through individual channel reaches and a time-dependent mixing model to estimate the concentration of E. coli bacteria in each reach.

 

The continuity equation for streamflow routing can be expressed as follows:

                                                                                                       (1)

where Q is the volumetric discharge (m³/s), Vw is the volume of water stored in the channel reach (m³), and t is time.  It is standard engineering practice to assume a simple linear relationship between the storage and flow so that flow routing can be accomplished using the Muskingum equation (McCarthy, 1938):

                                                                         (2)

where the subscript t refers to the current calculation time-step and t-1 refers to the previous time-step.  In this equation, Q_in includes the inflows from all tributaries to the reach as well as the lateral inflows (runoff) from the adjacent interfluves (hillslopes):

                                                                                                   (3)

where Q_u1 and Q_u2 are the discharges at the lower ends of the two stream segments that join to form the internal trunk stream segment, and R_i is the volumetric runoff from the adjacent interfluves, which is computed from:

                                                                                                                (4)

where P is the rainfall rate, C_r is the weighted average runoff coefficient of the contributing interfluve surface (see Element 2 for derivation of runoff coefficient.), and A_i is the area of the contributing interfluve surface.  Note that for headwaters streams, Q_in = R_i because there are no inflowing tributaries to those stream segments.   

The coefficients C₀ – C₂ in Equation 2 are defined as follows:

                                                                              (5a)

                                                                              (5b)

                                                                              (5c)

where Δt is the calculation time step (60 seconds), X is a weighting factor that is normally set equal to 0.2, and K is a storage coefficient that depends on the characteristics of the specific segment (for example, length, slope).

 

The mixing equation used in the hydrologic model is expressed as follows:

                  (6)

where Ec_r is the average E. coli concentration of the stream segment, Ec_in is the E. coli concentration of the tributary inflows, and Ec_i is the E. coli concentration of the runoff  from adjacent interfluves.

 

A computer program was written to simultaneously solve all of the equations above in a consistent and efficient manner. The method using algorithm described above has the following useful characteristics.

1. Each link (channel segment) and each node (channel junction) is assigned a unique code.

2. External links are distinguishable from internal links on the basis of their codes.  

3. Nodes that exist at both the upstream and downstream ends of each link within the network are readily accessible.

4. The method is applicable to virtually any stream channel network, and it is readily programmable.

 

The simulations conducted as part of this project could be executed in less than 30 seconds. Owing to this computational efficiency and the model's direct linkage to data of the Lake Michigan Rim GIS, it is feasible to use this method to study the effects of individual source areas by changing their land-use characteristics (for example, from non-sewered to sewered residential areas) and re-running the model. Differences between individual model runs become a quantitative measure of the improvement in water quality resulting from the specific land-use change.        

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