123 



In two-dimensional flow, the Laplace equation is also satisfied in terms of 

 the stream function, ^, so that a field of 4> and V can be constructed to 

 represent the usual flow net with lines of constant <^ representing lines of 

 equal head and lines of constant ip being streamlines and everywhere 

 perpendicular to lines of <^. 



With the governing equations noted above, many solutions to practical 

 problems can be obtained through conformal mapping procedures. For 

 particular geometries, possibly time-varying boundary conditions, and 

 complicated flow boundaries, numerical solutions are applied. The U.S. 

 Geological Survey developed and made available (McDonald and Harbaugh, 

 1984) a numerical model to simulate groundwater flow in three dimensions. 



8.3.2 Discharge through an Unconfined Aquifer 



Kozeny (1953) has presented a solution which has been modifed for flow 

 conditions from an unconfined aquifer to a shoreline. The solution is 

 developed in terms of complex variables and a distinct interface between 

 salt and fresh water and yields several results of interest. Denoting x 

 and y as the inland and downward coordinates (see Fig. 8.4), respectively, 

 the interface between fresh and sea water is expressed by 



2Q^ + -QJ- (8.8) 



7K -y'^K^ 



in which Q = freshwater flow per unit length of shoreline, and 7 = 

 (ps - Pf)/pf. The horizontal width, Xq, through which the fresh water 

 flows to the sea is 



o 27K 



The variation of freshwater head, h, with distance from the shoreline 

 is 



(8.10) 



