The coordinate system has been oriented so that 

 the y direction is vertical and parallel to the 

 direction of gravity. In equations (2) we assume 

 that permeability is the same in every direction. 

 Equations (2) imply the single vector equation 



or 



M L 9c J 



(3) 



where V is the gradient operator defined by 



V=i^^ — |-j ^^ — hk >r- and i, j, and k are unit vectors 

 da; •* dy oz ^ 



respectively in the three mutually perpendicidar 



directions of the distance variables x, y, and s. 



For later application of Darcy's law it is con- 



venient to define the expression (f>-= 

 WTite equation (3) as 



.icP 



gp 



+2/ and re- 



(4) 



Note that 0, the potential for intragravel flow, 

 represents the total energy of intragi'avel water 

 and is the sum of pressm-e energy and elevation 

 energy. Kinetic energy is assumed to be negligible. 

 The continuity equation or equation of conser- 

 vation of mass for an incompressible fluid is 



V-v=0 



(5) 



The substitution of Darcy's law, equation (4), 

 into equation (5) gives 



kV~4>+{yk)-V<i>=0 



(6) 



If the permeability, k, is homogeneous or uniform 

 (i.e., independent of position) equation (6) reduces 

 to Laplace's equation. 





(7) 



Meaningful solutions of equation (6) or (7) can- 

 not be found until the boundary conditions (i.e., 

 the values of or derivatives of 4> along the 

 boundaries of the porous bed) are specified. 



I have thus far discussed any laminar flow 

 described by Darcy's law. Components of water- 



flow within a streambed are shown in figure 2. 

 The porous streambed, .4, is considered to be a 

 stable bed consisting of particles and having a uni- 

 form permeability, k. The ambient stream, B, is 

 continuous and uniform. The porous streambed 

 is bounded below by an impermeable stratum, C. 

 Any flow, D, within the porous bed is termed in- 

 tragravel flow and is characterized by a velocity 

 vector, V. Any flow across the upper boundary of 

 the bed is termed interchange. An upward inter- 

 change from the bed to the stream is upwelling, 

 F; a downward interchange is downweUing, E. 



The above conditions of uniform permeability 

 and an impermeable stratum, for which equation 

 (7) applies, characterize most manmade spawning 

 channels. The permeability of natural streambeds 

 usually decreases with depth, and flow is described 

 by equation (6). Where permeability varies with 

 depth, one may measure permeability at several 

 depths and assign an appropriate permeability 

 function, k{y). 



Equation (6) or equation (7) for uniform per- 

 meability characterizes aU laminar flow through 

 porous materials; the boundary conditions dis- 

 tinguish particidar seepage situations. Our con- 

 sideration of intragravel flow is limited to a 

 two-dimensional system for which four boundary 

 conditions must be specified about a bed of finite 

 dimensions. An infinitely long bed requires speci- 

 fication of two boundary conditions. Because there 

 is no flow across the lower boundary of the porous 

 bed, the component of waterflow normal to the 

 boundary at the boundary is and 



d0 



dn, 



boundary, i/2(x)=0 



(8a) 



CONVEX STREAM 

 — PROFILE 



Figure 2. — Description of terms and properties relating 

 to waterflow within a streambed. A, porous bed; B, 

 ambient stream; C, impermeable stratum; D, intragravel 

 flow; E, downweUing; F, upwelling; y^, bed surface pro- 

 file; j/2. impermeable stratum profile. 



INTRAGRAVEL FLOW OF WATER IN A STREAMBED 



481 



