where k is the slope of the line of best fit of the semilog plot of S 

 versus t, f^ is the initial fill excursion, C is the fraction of 

 f^ lost after initial losses (i.e., at t=t£), and S t is the 

 excursion loss at time t due to sorting and slope adjustments of a 

 beach fill. 



Note that the exponential form of equation (4) implies that the 

 initial losses, although very small, continue indefinitely. However, 

 the excursion plots indicate that after 1 to 2 years the excursion loss 

 due to slope adjustment and initial sorting cannot be separated from the 

 seasonal and long-term losses. Hence, for practical reasons, the 

 initial loss will be mathematically considered complete when 95 percent 

 of£f£ is lost (i.e., at t=t£). 



The total excursion loss, D- , at time t after fill placement, is 

 the sum of equations (3) and (4). 



D = C f-(l-10" kt ) + at (5) 



t ^ l 



or, the total beach excursion relative to the prefill position, E t , 

 at any time t after a fill, is 



., ■ ■• [- 



E = f. l-£+ £io 



-kt 



- at (6) 



Equation (6) is an important tool which can be used to evaluate 

 historic beach fills and to design future ones. This equation -can be 

 used in two ways. First, if a given design lifetime of a fill is 

 required, substituting E t =0 and t equal to the desired design life, 

 then equation (6) is solved to give the initial fill excursion (and 

 volume). Second, for a given volume of fill, or alternatively, for a 

 given initial excursion, the time t=t e at which the beach returns to 

 its prefill position (E t =0) can be determined (i.e., the "useful 

 life" of the fill can be determined). These calculations can be used to 

 quantify the effectiveness and value of a given beach fill. However, 

 the assumption made within these interpretations of equation (6) is that 

 the beach fill has lost its effectiveness as soon as the beach face 

 between the MLW to MHW contours returns to its initial, prefill 

 position. It must be noted that in addition to providing a horizontal 

 excursion of the beach face, beach fills provide, either directly or 

 indirectly, three other functions which retain their value even when the 

 initial excursion is lost. The direct value is that the elevation of 

 the berm(s) and sometimes dunes is increased during beach-fill opera- 

 tions so that a larger volume of material seaward of the backdune is 

 available to absorb the erosional tendencies of storm waves. This pro- 

 vides an additional degree of protection to the backshore which was not 

 present prior to the fill placement. Indirectly, beach fills result in 

 an increase in sand on downdrift beaches, and produce slight decreases 

 in the nearshore to offshore bathymetry due to the redistribution of 

 beach-fill material offshore as a result of slope readjustment and 



44 



