APPENDIX B: EQUATIONS USED TO CALCULATE DISCHARGE.
Vertical Discharge 

The vertical flux, Q_v, for the control volume in Figure 3.7 was calculated using Darcy's law and the head data from the piezometers situated 3 m from the creekbank at depths of 50 cm and 75 cm. When the head at 50 cm is greater than 75 cm, there is a downward flux into the control volume, and when the reverse is true, there is an upward flux into the control volume. Because both upward and downward fluxes represent flow into the control volume, Q_v is taken to be always positive. The equation used to calculate the vertical flux is:

(B-1)  

where Q_v is the flux in liters min^-1 at time t_n, K is the saturated hydraulic conductivity, h₇₅ and h₅₀ are the measured hydraulic heads at the 75 cm and 50 cm deep piezometers, z₇₅ and z₅₀ are the actual depths of the piezometers (z is positive in the downward direction), and x and y are the dimensions of the control volume in the x and y directions. x and y are both set at 100 cm. The instantaneous discharge for each time step in the data set (time step = 12 min) was obtained using the finite difference form of equation (B1):

(B-2)  

Q_v has units of L min^-1. The mean Q_v for each stage (12 minute interval) was determined by taking all discharge estimates with the same stage and averaging. This results in 63 sequential estimates of instantaneous discharge. The total upward and downward vertical discharges were determined by taking the integral over the total time interval where Q_v represents a flux up or a flux down, such that:

(B-3a) , for all h₇₅ > h₅₀ 

(B-3b) , for all h₇₅< h₅₀ 

The total Q_v, in units of L m^-2 (tidal cycle)^-1 are determined by summing Q_v-up and Q_v-down.

Horizontal discharge 

The horizontal discharge values were estimated as shown for the control volume in Figure 3.7. For each pair of piezometers, BA, CB and DC the specific discharge was calculated using the following equations:

(B-4a)  

(B-4b)  

(B-4c)  

In each of these equations, a negative discharge indicates flow towards the creekbank, and a positive discharge indicates flow towards the interior of the marsh. To estimate the flow in and out of the yz faces of the control volume, the average discharge from above and below the control volume was averaged with the flow through the control volume. The assumption has been made that the averaged flow obtained using this method is representative of flow through the region defined by piezometers ABC or BCD. The equations describing these instantaneous fluxes are:

(B-5a)  

(B-5b)  

The horizontal discharge for each point in the tidal cycle, Q was calculated in a similar fashion as the vertical discharge. The volume of the flux was estimated using an area of the face defined by y and the height of the face at that point in time, h_C or h_B. In doing this, only flow through the saturated portion of the control volume was taken into account. When the height of water in the piezometer was greater than the surface of the marsh (i.e. the surface is flooded) the height of the control volume was equal to the depth from MLW to the surface of the sediment. In finite difference form, the instantaneous flux through each yz face is:

(B-6a)  

(B-6b)  

The mean flux for each tidal stage was calculated, and the fluxes into and out of the upper and lower yz faces were calculated by integrating the positive and negative values of the discharge separately, as was done for the vertical fluxes, such that:

(B-7a) , for all Q_DCB < 0,

(B-7b) , for all Q_DCB > 0,

(B-7c) , for all Q_CBA > 0,

(B-7d) , for all Q_CBA < 0.

The total horizontal discharge in (Q_h - in) and out (Q_h - out) of each face was determined by summing the fluxes into and out of the control volume. The quantities have units of L m^-2 (tidal cycle)^-1.

Total Discharge 

The total discharge in to and out of the control volume, in units of L m^-2 (tidal cycle)^-1 was determined using the following equations:

(B-8a)   

(B-8b)  

where ET is the flux out of the surface of the marsh due to evapotranspiration. The water balance was achieved by assuming that total_in and total_out are equal, and that the difference between total_in and total_out is equal to the error associated with the measurements.


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