negative slope and is convex if 0<ip<<t>{0)/ir, and 

 concave if O>p>-0(O)/t (See figure 2). The 

 solution to the boundary value problem equation 

 (8a-d) for this surface potential and a bed of 

 uniform permeability is 



, cosh (Try/a) . / , v . ,/o\ <^(0)^ 



(10) 



If we define interchange as the vertical (y- 

 directed) component of intragravel flow at the 

 streambed surface (fig. 2) the exact magnitude of 

 interchange is given by 



J pg i)<t> . 



(11) 



For the example at hand the interchange velocity 



This example shows that the direction of inter- 

 change depends on the shape of the streambed 

 surface. A convex streambed profile for which p 

 is positive induces a negative interchange velocity 

 or downweUing; a concave profile for which p is 

 negative causes upwelling. 



2. Statement of the boundary value problem 

 for an infinitely long streambed requires only 

 equations (6), (8a), and (8d). We may solve the 

 boundary value problem for the flow within 

 an infinitely long streambed by the mathe- 

 matics of complex variables (Weinberger, 1965, 

 pp. 201-268). The transformation f = (cosh Z— 1)/ 

 sinh Z conformally maps an infinitely long 

 streambed of depth b = ir/2 in the Z plane into a 

 half disk in the f plane. The boundary potential 

 </)(a;) = — g tanh x for — co<x<oo where g is a 

 positive dimensionless, adjustable parameter is 

 associated with the sigmoid-shaped streambed in 

 figure 2. The solution of the boundary value 

 problem is <\>{x,y) = —ci sinh a;/ (cosh x+cos y). 

 From equation (11) we find the interchange 

 velocity to be 



V»|y=6 = A:2 



P9 



sinh X . 



y. cosh^ X 



J 



(13) 



As in the previous example, downwelling takes 

 place in a convex streambed profile (— oo^x<^0) 

 and upwelling in a concave profile (0<a:<o°). 



ANALOG MODEL INVESTIGATION OF 

 INIRAGRAVEL FLOW 



An electrolytic-bath analog model was used to 

 solve the boundary value problem (equations 7 

 and 8a-8d) for several specified boundary condi- 

 tions and » streambed of uniform permeabiUty. 

 This technique of solving the partial differential 

 equations of intragravel flow rests upon the anal- 

 ogy between Darcy's and Ohm's laws and the 

 respective steady-state conservation equations of 

 mass and electrical charge. An electrolytic equiva- 

 lent of a streambed section with intragravel flow 

 can be prepared by constructing a shallow tray 

 geometrically similar to the two-dimensional por- 

 ous bed. The depth of the conducting liquid at 

 any point in the tray is proportional to the per- 

 meability at the corresponding point in the porous 

 bed. The conducting tray walls, usually metal 

 strips, are set at voltages proportional to the 

 analogous hydrodynamic flow potentials or values 

 of 4, (Zangar, 1953, p. 24). 



EQUIPMENT AND PROCEDURE 



The electrolytic-bath model constructed for this 

 investigation was a 10- by 20-cm. plexiglass tray 

 filled to a depth of 2.5 cm. with, a 0.01 normal 

 solution of aqueous sodium chloride (fig. 4). The 

 end walls representing constant potential (con- 

 stant 0) surfaces were brass strips; a nonconduct- 

 ing strip represented a water-impermeable surface. 

 Rheostat-controlled voltages applied separately to 

 each of 15 electrodes along the variable-potential 

 waU {y=h) provided an approximation to the 

 potential function, <t>{x). 



In all operations of the analog model, 10 volts 

 of 60-cycle alternating current were impressed 



NONCONDUCTING STRIP 



ELECTRODES 



eRASS STRIP 



NONCONDUCTING STRIP 



Figure 4. — Electrical circuits and physical layout for 

 electrolytic-bath analog model used to investigate flow 

 of intragravel water in a streambed. 



INTRAGRAVEL FLOW OP WATER IN A STREAMBED 



483 



