where n is the direction normal to the impermeable 

 surface. 



The velocity vector at any point within a 

 streambed is influenced by streambed geometry 

 and boundary conditions over the length of the 

 stream. For investigation of intragravel flow in a 

 finite reach of stream, between a;=0 and x=a, we 

 ignore upstream and downstream conditions. The 

 boundary conditions 0(0,?/) and 4>{a,y) depend 

 upon conditions over all the streambed. Since 

 0(x, yi{x)), hereafter denoted by <\>{x), is unknown 

 outside of 0<x<a, it is necessary to assign 

 reasonable estimates of <^ along the upstream and 

 downstream boundaries of the isolated region. 

 Accordingly, in isolating a region of streambed of 

 depth 6 between a;=0 and x=a, I specify the 

 boundary conditions 0(O,i/) and 4>{a,y) to be con- 

 stant and respectively equal to <t>{Q,h) and <t>{a,b). 

 The physical model is shown in figiu"e 3(i) 



^{0,y) = <p{0,b) (8b) 



4>ia,y) = <l>{a,b) (8c) 



Because of continuity of pressure across the 

 stream-gravel interface, the potential at the upper 

 bed boundary, 4>[x, yi(x)] = <pix), must be equal to 

 the potential at the bottom of the contiguous 

 flowing stream. The problem of determining the 

 potential along the upper boundary, 4>{x), then 

 becomes one of hydraulics. 



The bottom pressure, P{x,yi), of a flowing 

 stream is related to properties of the stream by 



Pix,y:)=^d{x)+Po+Pc{x,y,) (9) 



ye 



« = *(n) 



<> = <>I01 



«= « 1k1 



«=*(ol 







a* 



Figure 3. — (i) Boundary value problem representing the 

 physical model for flow through a streambed of porous 

 material. The shape of the region is the actual shape of 

 the porous bed in which the upper and lower boundary 

 curves are the topographical profiles of the bed surface 

 and impermeable layer, (ii) Part (i) mapped into a 

 rectangle. 



in which d(x) is the stream depth, P„ is atmos- ! 

 pheric pressure, and Pc is centrifugal pressure due 

 to stream curvature. From a stream-(-ontinuity 

 equation, d{x)U{x)= constant, and the Manning 

 flow formula,' U{x)d{x)-''^ 2/'l(a;)-"'=^constant, 

 where y' is the hydraulic gradient and U is the 

 average stream velocity, I find that the stream 

 depth varies with the 0..3 power of hydraulic 

 gradient. Considering small changes in stream 

 gradient so that changes in Pdx) and d{x) are 

 smaU, it follows that changes in the right-hand 

 side of equation (9) are small compared with 

 changes in y'{x) and that gcP /gp^^const&nt, from 

 which we may appro.ximate the potential of the 

 upper bed boundary 



,^(z)= 2/1 (x)+^^^^^^M-;i/,(x)-h constant (8d) 

 9P 



The applicability of equation (8d) is limited to 

 reaches of nearly constant stream depth (see 

 footnote 3). Because the average velocity of 

 streamflovv is several orders of magnitude greater 

 than the velocity of intragravel flow, the influence 

 of interchange on stream discharge is negligible. 



Equation (6) or equation (7) for uniform 

 permeability and the equations for the four 

 boundary conditions (8a-8d) complete the bound- 

 ary value problem. Figure 3(i) depicts the model 

 that is considered for particular solutions. Solution 

 of the boundary value problem is facilitated by 

 assuming the bed to be of constant depth, b (so 

 that yi = yi-\-b (fig. 2)) and by mapping the 

 irregularly bounded region into a rectangle, as in 

 figure 3(ii). The error introduced is negligible 

 because, over short distances (e.g., less than 10 m.), 

 natural streambeds commonly have almost con- 

 stant depth and are nearly horizontal. Boundary 

 potentials are unchanged by formation of a 

 rectangle. 



Analytical solutions of the boundary value 

 problem describing intragravel flow usually involve 

 the use of uifinite series, i.e., are of open form. 

 Two simple closed-form solutions arise in the 

 methods used here to solve the problem. 



1. A potential <^(a:) = </>(0) — (/)(0)z/a+2'*''n (irx/a) 

 in which p is an arbitrary constant holds in a 

 curved streambed surface profile for 0<a;<[a. 

 According to equation (8d) the profile has a 



' This (orm of the Mannhig equation implies that the stream hydraulic 

 radius Is equal to d(j), that bed roughness Is constant In the direction of flow, 

 and that the stream energy gradient is parallel to the streambed surface. 



4^ 



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