of Chemical Engineering, University of Minne- 

 sota (Vaux, 1961). 



All symbols are defined the first time they ap- 

 pear and are listed with definitions and units in 

 the "Notation" section near the end of this paper. 



THEORY OF INTRAGRAVEL 

 WATERFLOW 



Charles Slichter (1899) first applied the mathe- 

 matics of partial differential equations to the mo- 

 tion of groundwater (waterflow in saturated soil). 

 His formulation of the ecjuations describing 

 groundwater flow provided a foundation for sub- 

 sequent analyses of similar problems of intragravel 

 waterflow. 



In developing the theory of intragravel water- 

 flow, I treat the case where the rate equation for 

 flow in porous media is linear, i.e., conditions 

 under which Darcy's law applies. Ergun (1952) 

 showed that for a given particle size and bed 

 porosity, the ratio of pressure gradient to water 

 velocity in a porous bed is given by (AP/L)/ 

 V a (fii-\-C2Ng^ where Ci and c, are known con- 

 stants, and Nrc is the Reynolds number based on 

 particle diameter. The constants Ci and C2 respec- 

 tively account for pressure loss from the effects 

 of viscous and kinetic energy. The expression 

 (AP/L)/v oc c, caUed Darcy's law, is considered 

 applicable for Reynolds numbers less than unity 

 (Rumer, 1965). Although flow may be laminar in 

 situations where the Reynolds numbers are higher 

 than unity, the effects of convective accelera- 

 tion in such situations invalidate Darcy's law 

 (Silberman, 1965). 



Analysis of data on gravel size, permeability, 

 and hydraulic gradient from the Carmen-Smith 

 spawning channel,' and Jones Creek, Alaska, 

 spawnmg channels, McNary, Priest Rapids, and 

 Robertson Creek,^ and my data from Sashin 

 Creek, Alaska, indicates that intragravel flow is 

 of the hnear laminar type and, therefore, described 

 by Darcy's law. I assume that these cases describe 

 the usual conditions in salmon spawning beds 

 and that Darcy's law is generally apphcable in 

 the description of intragravel flow. 



For situations in which the Reynolds numbers 

 are less than unity, the velocity of laminar seepage 



' Hagey, Dale W., and Robert T. Ounsolus, Progress report on the oper- 

 ation and evaluation of the Cannen-Smith spawning channel, 1960-64. 

 Oregon Fish Commission, Feb. 196S, 12 pp. 



» Data supplied by the Columbia Fisheries Program Office, Bureau of 

 Commercial Fisheries, Portland, Oreg. 



flow within a porous medium, v, is related to the 

 pressure gradient, AP/L by 



.^^i.9c^ 





(1) 



This relation is termed Darcy's law, but rather 

 than a law, it is actually an equation wliicli de- 

 fines k, the "specific permeability," or just "per- 

 meabihty." In equation (1), n is the liquid vis- 

 cosity, and Qc is the constant of Newton's second 

 law, introduced to make the equation dimension- 

 ally correct (g, equals 1 g.-cm./dyne-sec.^ in 

 centimeter-gram-second units) . 



The value of AP can be calculated by the 



relation ^P=P^ Mi. where p is liquid density, 



Ah is head loss over the distance L, and g is 

 acceleration due to gravity. Figure 1 illustrates 

 the variables v, k, Ah, and L. The head loss. Ah, 

 is shown in figure 1 by the elevations to which 

 water rises in the piezometer tubes at opposite 

 ends of the column of porous material. 



We may extend Darcy's law to account for flow 

 due to gravity, in addition to pressure, and write 

 expressions for velocity in each of the three 

 directions of a cartesian coordinate system. Ratios 

 for AP/L can be replaced with the corresponding 

 derivatives : 



fi ox 





vr- 



n oz 



(2a) 

 (26) 

 (2c) 



--. l?5i;(,-.aP0RbuS" MATERIAL 6f<§?;.J ~- S § 

 --l*50.°c*7< PERMEABILITY kMS.O'y-- 5§ 



Figure 1. — Rectilinear waterflow through a sample of 



porous material showing a head loss Ah = AP - over a 



9P 

 distance L. 



480 



U.S. FISH AND WILDLIFE SERVICE 



